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Properties

Label 4-961311-1.1-c1e2-0-4
Degree $4$
Conductor $961311$
Sign $-1$
Analytic cond. $61.2940$
Root an. cond. $2.79804$
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  + 3·3-s − 4-s + 4·9-s + 3·11-s − 3·12-s − 13-s − 3·16-s − 8·17-s + 4·19-s + 25-s − 31-s + 9·33-s − 4·36-s + 37-s − 3·39-s − 3·44-s − 19·47-s − 9·48-s + 5·49-s − 24·51-s + 52-s + 12·57-s + 7·64-s − 2·67-s + 8·68-s − 19·71-s + 3·75-s + ⋯
L(s)  = 1  + 1.73·3-s − 1/2·4-s + 4/3·9-s + 0.904·11-s − 0.866·12-s − 0.277·13-s − 3/4·16-s − 1.94·17-s + 0.917·19-s + 1/5·25-s − 0.179·31-s + 1.56·33-s − 2/3·36-s + 0.164·37-s − 0.480·39-s − 0.452·44-s − 2.77·47-s − 1.29·48-s + 5/7·49-s − 3.36·51-s + 0.138·52-s + 1.58·57-s + 7/8·64-s − 0.244·67-s + 0.970·68-s − 2.25·71-s + 0.346·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961311 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961311 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(961311\)    =    \(3 \cdot 13 \cdot 157^{2}\)
Sign: $-1$
Analytic conductor: \(61.2940\)
Root analytic conductor: \(2.79804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 961311,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
157$C_2$ \( 1 + 25 T + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ad_m
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.i_bu
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.23.a_p
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.29.a_ba
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.b_ak
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ab_cq
41$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.41.a_acb
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.t_gw
53$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.53.a_c
59$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.59.a_acy
61$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.61.a_acb
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.c_ep
71$C_2$$\times$$C_2$ \( ( 1 + 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.t_iy
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.79.a_acy
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.89.c_acz
97$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.97.a_be
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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

−8.150467818171834488610710637020, −7.48687961828468267309351189874, −7.16740500706871057543377867413, −6.74688839072444554024791577204, −6.30047493765635272108561055297, −5.70762301987707015806254310957, −4.93608646533576050030667301286, −4.61081159001824922269620635930, −4.12718767561803680872179162093, −3.68353641398139212057840080491, −3.07529552861539263039694233706, −2.63768646049488002186868027872, −2.01826907609250944288540364827, −1.43562235377481692507206635595, 0, 1.43562235377481692507206635595, 2.01826907609250944288540364827, 2.63768646049488002186868027872, 3.07529552861539263039694233706, 3.68353641398139212057840080491, 4.12718767561803680872179162093, 4.61081159001824922269620635930, 4.93608646533576050030667301286, 5.70762301987707015806254310957, 6.30047493765635272108561055297, 6.74688839072444554024791577204, 7.16740500706871057543377867413, 7.48687961828468267309351189874, 8.150467818171834488610710637020

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