Properties

Label 4-1815e2-1.1-c3e2-0-4
Degree $4$
Conductor $3294225$
Sign $1$
Analytic cond. $11467.9$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 4-s − 10·5-s + 27·9-s − 6·12-s − 60·15-s − 63·16-s + 10·20-s + 96·23-s + 75·25-s + 108·27-s + 320·31-s − 27·36-s − 532·37-s − 270·45-s − 1.00e3·47-s − 378·48-s − 446·49-s − 684·53-s − 1.32e3·59-s + 60·60-s + 127·64-s + 992·67-s + 576·69-s − 1.41e3·71-s + 450·75-s + 630·80-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/8·4-s − 0.894·5-s + 9-s − 0.144·12-s − 1.03·15-s − 0.984·16-s + 0.111·20-s + 0.870·23-s + 3/5·25-s + 0.769·27-s + 1.85·31-s − 1/8·36-s − 2.36·37-s − 0.894·45-s − 3.12·47-s − 1.13·48-s − 1.30·49-s − 1.77·53-s − 2.91·59-s + 0.129·60-s + 0.248·64-s + 1.80·67-s + 1.00·69-s − 2.36·71-s + 0.692·75-s + 0.880·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3294225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(11467.9\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3294225,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 - p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{6} T^{4} \)
7$C_2^2$ \( 1 + 446 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 - 466 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9586 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 13658 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )^{2} \)
29$C_2^2$ \( 1 + 5038 T^{2} + p^{6} T^{4} \)
31$C_2$ \( ( 1 - 160 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 266 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 106102 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 + 63014 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 504 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 + 342 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 + 660 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 406922 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 496 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 708 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 364694 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 954338 T^{2} + p^{6} T^{4} \)
83$C_2^2$ \( 1 + 1136314 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 606 T + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 254 T + p^{3} T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.588162672267354936735080070137, −8.377101091443660710738583181545, −8.015174743625218510657658139800, −7.58980685279061000693114662684, −7.30715583388747158873968004724, −6.60918771942042557881963038122, −6.58958265000278309963838252316, −6.14472012216150194959722550708, −5.08734087938487562879417624664, −4.96316912492890356884526983238, −4.54334867391187319414925652661, −4.20648126481112359014036648408, −3.40905520143395471332860680440, −3.11281506120626705124150229800, −3.05049779434701046353107108311, −2.12867523949212963342903612660, −1.62595026429531559448008595461, −1.16494356637207270430693701809, 0, 0, 1.16494356637207270430693701809, 1.62595026429531559448008595461, 2.12867523949212963342903612660, 3.05049779434701046353107108311, 3.11281506120626705124150229800, 3.40905520143395471332860680440, 4.20648126481112359014036648408, 4.54334867391187319414925652661, 4.96316912492890356884526983238, 5.08734087938487562879417624664, 6.14472012216150194959722550708, 6.58958265000278309963838252316, 6.60918771942042557881963038122, 7.30715583388747158873968004724, 7.58980685279061000693114662684, 8.015174743625218510657658139800, 8.377101091443660710738583181545, 8.588162672267354936735080070137

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