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Properties

Label 4-104e2-1.1-c1e2-0-2
Degree $4$
Conductor $10816$
Sign $1$
Analytic cond. $0.689637$
Root an. cond. $0.911287$
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  − 2·2-s + 2·4-s + 6·7-s + 5·9-s − 12·14-s − 4·16-s − 14·17-s − 10·18-s + 8·23-s + 25-s + 12·28-s − 16·31-s + 8·32-s + 28·34-s + 10·36-s + 4·41-s − 16·46-s − 14·47-s + 13·49-s − 2·50-s + 32·62-s + 30·63-s − 8·64-s − 28·68-s − 6·71-s + 28·73-s − 20·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 2.26·7-s + 5/3·9-s − 3.20·14-s − 16-s − 3.39·17-s − 2.35·18-s + 1.66·23-s + 1/5·25-s + 2.26·28-s − 2.87·31-s + 1.41·32-s + 4.80·34-s + 5/3·36-s + 0.624·41-s − 2.35·46-s − 2.04·47-s + 13/7·49-s − 0.282·50-s + 4.06·62-s + 3.77·63-s − 64-s − 3.39·68-s − 0.712·71-s + 3.27·73-s − 2.25·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10816\)    =    \(2^{6} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.689637\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6630077108\)
\(L(\frac12)\) \(\approx\) \(0.6630077108\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.17.o_df
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.43.a_adh
47$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.47.o_fn
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.59.a_da
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.71.g_fv
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.83.a_be
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−14.15440631000278160403879527824, −13.20852039438558746320617634653, −13.19648486565889657619060078782, −12.58600300582480593983015104864, −11.34323865242704371941958808037, −11.26399418836677676300121220421, −10.92856591718551450502520099306, −10.52597930620038220776880637397, −9.508812300858634453042338551359, −9.199662015815668273650387336959, −8.626171664997409426871552911051, −8.195138470375305827482347337470, −7.40053747850704823055861792041, −7.10015539971666652880095457705, −6.57035023944590742165974568824, −5.01634651797727751766763907520, −4.73879432561462994763240805629, −4.10530874863142447259873660113, −2.05759140072052388639569224374, −1.61690266206879990138128123661, 1.61690266206879990138128123661, 2.05759140072052388639569224374, 4.10530874863142447259873660113, 4.73879432561462994763240805629, 5.01634651797727751766763907520, 6.57035023944590742165974568824, 7.10015539971666652880095457705, 7.40053747850704823055861792041, 8.195138470375305827482347337470, 8.626171664997409426871552911051, 9.199662015815668273650387336959, 9.508812300858634453042338551359, 10.52597930620038220776880637397, 10.92856591718551450502520099306, 11.26399418836677676300121220421, 11.34323865242704371941958808037, 12.58600300582480593983015104864, 13.19648486565889657619060078782, 13.20852039438558746320617634653, 14.15440631000278160403879527824

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