Properties

Label 8-55e4-1.1-c9e4-0-0
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $643873.$
Root an. cond. $5.32230$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 272·3-s + 3.69e4·9-s − 5.24e5·16-s − 5.36e6·23-s + 3.06e6·25-s + 3.22e5·27-s + 3.15e7·37-s + 2.23e7·47-s + 1.42e8·48-s + 1.44e8·53-s + 6.06e8·67-s + 1.45e9·69-s − 1.71e9·71-s − 8.33e8·75-s − 1.11e9·81-s − 1.86e8·97-s + 4.46e9·103-s − 8.59e9·111-s − 1.11e9·113-s + 4.71e9·121-s + 127-s + 131-s + 137-s + 139-s − 6.07e9·141-s − 1.93e10·144-s + 149-s + ⋯
L(s)  = 1  − 1.93·3-s + 1.87·9-s − 2·16-s − 3.99·23-s + 1.56·25-s + 0.116·27-s + 2.77·37-s + 0.667·47-s + 3.87·48-s + 2.51·53-s + 3.67·67-s + 7.74·69-s − 7.99·71-s − 3.04·75-s − 2.86·81-s − 0.213·97-s + 3.90·103-s − 5.37·111-s − 0.641·113-s + 2·121-s − 1.29·141-s − 3.75·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(643873.\)
Root analytic conductor: \(5.32230\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :9/2, 9/2, 9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.4943551757\)
\(L(\frac12)\) \(\approx\) \(0.4943551757\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_2^2$ \( 1 - 3063526 T^{2} + p^{18} T^{4} \)
11$C_2$ \( ( 1 - p^{9} T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p^{5} T + p^{9} T^{2} )^{2}( 1 + p^{5} T + p^{9} T^{2} )^{2} \)
3$C_2$$\times$$C_2^2$ \( ( 1 + 136 T + p^{9} T^{2} )^{2}( 1 - 20870 T^{2} + p^{18} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 2682396 T + p^{9} T^{2} )^{2}( 1 + 3592942977890 T^{2} + p^{18} T^{4} ) \)
29$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 26636854831058 T^{2} + p^{18} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 - 15796298 T + p^{9} T^{2} )^{2}( 1 - 10400449085350 T^{2} + p^{18} T^{4} ) \)
41$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 11166228 T + p^{9} T^{2} )^{2}( 1 - 2113576298457550 T^{2} + p^{18} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 - 72333594 T + p^{9} T^{2} )^{2}( 1 - 1367378362647430 T^{2} + p^{18} T^{4} ) \)
59$C_2$ \( ( 1 - 70370640 T + p^{9} T^{2} )^{2}( 1 + 70370640 T + p^{9} T^{2} )^{2} \)
61$C_2$ \( ( 1 - p^{9} T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 303360928 T + p^{9} T^{2} )^{2}( 1 + 37614783844431290 T^{2} + p^{18} T^{4} ) \)
71$C_2$ \( ( 1 + 427903668 T + p^{9} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p^{9} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{18} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 623295746335388018 T^{2} + p^{18} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 93053342 T + p^{9} T^{2} )^{2}( 1 - 1511803192851761470 T^{2} + p^{18} T^{4} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.695822929066144435143273079276, −8.877682922601475903965281111545, −8.728482506446159795119395807927, −8.661569459976989044774344336675, −8.068944701493273174504221137599, −7.61505417733310832977003813904, −7.32322133146230875767438695560, −7.00186519084016620136275588822, −6.68850399998916412501141706338, −6.22229130943377302667272308653, −5.97540908412047329492186238931, −5.83745813718077568389428258250, −5.54307877659068072401103082450, −4.83890585648357931856513358661, −4.47219975601259007418952673371, −4.38310359776592592152533697767, −4.17022338955685321698349256827, −3.45629484560354232067312367403, −2.79364310152935371464996777199, −2.36409099372627261869402376774, −2.12755734233111889152741015368, −1.50665707031350652042684007701, −0.902695720718914740684074946387, −0.61364186658529410948951343452, −0.17982166302239824515192697878, 0.17982166302239824515192697878, 0.61364186658529410948951343452, 0.902695720718914740684074946387, 1.50665707031350652042684007701, 2.12755734233111889152741015368, 2.36409099372627261869402376774, 2.79364310152935371464996777199, 3.45629484560354232067312367403, 4.17022338955685321698349256827, 4.38310359776592592152533697767, 4.47219975601259007418952673371, 4.83890585648357931856513358661, 5.54307877659068072401103082450, 5.83745813718077568389428258250, 5.97540908412047329492186238931, 6.22229130943377302667272308653, 6.68850399998916412501141706338, 7.00186519084016620136275588822, 7.32322133146230875767438695560, 7.61505417733310832977003813904, 8.068944701493273174504221137599, 8.661569459976989044774344336675, 8.728482506446159795119395807927, 8.877682922601475903965281111545, 9.695822929066144435143273079276

Graph of the $Z$-function along the critical line