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Properties

Label 4-18e4-1.1-c1e2-0-3
Degree $4$
Conductor $104976$
Sign $1$
Analytic cond. $6.69336$
Root an. cond. $1.60846$
Motivic weight $1$
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  + 3·5-s − 2·7-s − 6·11-s − 5·13-s + 6·17-s + 4·19-s + 6·23-s + 5·25-s + 3·29-s + 4·31-s − 6·35-s + 10·37-s − 6·41-s + 10·43-s + 7·49-s + 12·53-s − 18·55-s − 12·59-s − 5·61-s − 15·65-s − 2·67-s − 12·71-s − 2·73-s + 12·77-s + 10·79-s + 18·85-s + 6·89-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.755·7-s − 1.80·11-s − 1.38·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 25-s + 0.557·29-s + 0.718·31-s − 1.01·35-s + 1.64·37-s − 0.937·41-s + 1.52·43-s + 49-s + 1.64·53-s − 2.42·55-s − 1.56·59-s − 0.640·61-s − 1.86·65-s − 0.244·67-s − 1.42·71-s − 0.234·73-s + 1.36·77-s + 1.12·79-s + 1.95·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(104976\)    =    \(2^{4} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(6.69336\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 104976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647586677\)
\(L(\frac12)\) \(\approx\) \(1.647586677\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_z
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.f_m
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_au
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ae_ap
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_cf
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_dh
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.73.c_fr
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.89.ag_hf
97$C_2^2$ \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_d
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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

−11.96297238117988667640298414736, −11.41470032453232554750095195641, −10.53374388284959934229764616553, −10.37864646771819563033577462290, −10.11664526213416518522555434750, −9.494133237303159615432348337833, −9.323598198158614980138775093847, −8.698233963198732900725570156511, −7.72041112268156637662184917747, −7.70411241119005063996128701604, −7.14427200571498360767566607545, −6.40285866834164375620609452158, −5.86595632836788444094026650577, −5.40080429617027325624458101214, −5.06591642150914633137430871306, −4.42296694578054834532579966763, −3.13604724997641892669080831300, −2.84814439494514772339290462718, −2.31321220030172474812161336887, −0.957563683851807472073539154453, 0.957563683851807472073539154453, 2.31321220030172474812161336887, 2.84814439494514772339290462718, 3.13604724997641892669080831300, 4.42296694578054834532579966763, 5.06591642150914633137430871306, 5.40080429617027325624458101214, 5.86595632836788444094026650577, 6.40285866834164375620609452158, 7.14427200571498360767566607545, 7.70411241119005063996128701604, 7.72041112268156637662184917747, 8.698233963198732900725570156511, 9.323598198158614980138775093847, 9.494133237303159615432348337833, 10.11664526213416518522555434750, 10.37864646771819563033577462290, 10.53374388284959934229764616553, 11.41470032453232554750095195641, 11.96297238117988667640298414736

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