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Properties

Label 4-31360-1.1-c1e2-0-9
Degree $4$
Conductor $31360$
Sign $-1$
Analytic cond. $1.99954$
Root an. cond. $1.18913$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·5-s − 3·7-s + 8-s − 5·9-s − 2·10-s − 11-s − 3·13-s − 3·14-s + 16-s − 5·18-s − 2·20-s − 22-s − 2·25-s − 3·26-s − 3·28-s + 31-s + 32-s + 6·35-s − 5·36-s − 2·40-s − 2·43-s − 44-s + 10·45-s − 6·47-s + 2·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s − 1.13·7-s + 0.353·8-s − 5/3·9-s − 0.632·10-s − 0.301·11-s − 0.832·13-s − 0.801·14-s + 1/4·16-s − 1.17·18-s − 0.447·20-s − 0.213·22-s − 2/5·25-s − 0.588·26-s − 0.566·28-s + 0.179·31-s + 0.176·32-s + 1.01·35-s − 5/6·36-s − 0.316·40-s − 0.304·43-s − 0.150·44-s + 1.49·45-s − 0.875·47-s + 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31360 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(31360\)    =    \(2^{7} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.99954\)
Root analytic conductor: \(1.18913\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 31360,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + T + p T^{2} ) \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.b_q
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.d_w
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.17.a_k
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.19.a_am
23$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.23.a_az
29$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.29.a_d
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ab_g
37$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.37.a_by
41$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \) 2.41.a_br
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.c_ck
47$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.g_da
53$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \) 2.53.a_ci
59$C_2^2$ \( 1 - 52 T^{2} + p^{2} T^{4} \) 2.59.a_aca
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.e_ej
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.71.a_i
73$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.73.a_ak
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.83.a_az
89$C_2^2$ \( 1 + 143 T^{2} + p^{2} T^{4} \) 2.89.a_fn
97$C_2^2$ \( 1 - 120 T^{2} + p^{2} T^{4} \) 2.97.a_aeq
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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

−10.30625158281814654646136601008, −9.662116873776763503704987997104, −9.290253827775280498435333499559, −8.389606765016871261693289332634, −8.163601736289340551621444198358, −7.49141551790521137416786289005, −6.81577031659550740336496387543, −6.38032383064270632520363812271, −5.60956005351140354594733538219, −5.25056001774719032595567424552, −4.35698877117903131179237458107, −3.60640018363373500720888002371, −3.04296613229166822081277949951, −2.40461908035731444788935104109, 0, 2.40461908035731444788935104109, 3.04296613229166822081277949951, 3.60640018363373500720888002371, 4.35698877117903131179237458107, 5.25056001774719032595567424552, 5.60956005351140354594733538219, 6.38032383064270632520363812271, 6.81577031659550740336496387543, 7.49141551790521137416786289005, 8.163601736289340551621444198358, 8.389606765016871261693289332634, 9.290253827775280498435333499559, 9.662116873776763503704987997104, 10.30625158281814654646136601008

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