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Properties

Label 4-961311-1.1-c1e2-0-2
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-s + 3·11-s − 6·13-s − 4·16-s + 3·17-s − 3·19-s + 3·25-s + 4·27-s − 6·31-s − 3·33-s + 3·37-s + 6·39-s − 12·47-s + 4·48-s − 4·49-s − 3·51-s + 3·57-s + 17·67-s + 24·71-s − 3·75-s − 7·81-s + 18·89-s + 6·93-s − 9·101-s + 9·109-s − 3·111-s + 9·113-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.904·11-s − 1.66·13-s − 16-s + 0.727·17-s − 0.688·19-s + 3/5·25-s + 0.769·27-s − 1.07·31-s − 0.522·33-s + 0.493·37-s + 0.960·39-s − 1.75·47-s + 0.577·48-s − 4/7·49-s − 0.420·51-s + 0.397·57-s + 2.07·67-s + 2.84·71-s − 0.346·75-s − 7/9·81-s + 1.90·89-s + 0.622·93-s − 0.895·101-s + 0.862·109-s − 0.284·111-s + 0.846·113-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$ \( ( 1 + T )( 1 + 5 T + p T^{2} ) \)
157$C_2$ \( 1 - 14 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ad_w
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ad_bi
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.d_bm
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.29.a_abe
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.g_ck
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.37.ad_ce
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.43.a_n
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.47.m_fa
53$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.53.a_cc
59$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.59.a_cr
61$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.61.a_dh
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.ar_he
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.73.a_cs
79$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \) 2.79.a_bx
83$C_2^2$ \( 1 + 99 T^{2} + p^{2} T^{4} \) 2.83.a_dv
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.as_jq
97$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.97.a_fs
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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

−7.935322747816453095042923623228, −7.43041297806588240718290428045, −6.88886624979378547960752879584, −6.53502042577489063110653387067, −6.43277175562229230913123730651, −5.57099709901224003127203453141, −5.11339976412187934944202631996, −4.87696576807597648427388114522, −4.33941267588924836802297779070, −3.71784894557919390235888777459, −3.20467365999786331932186000044, −2.38817897782664356706357578105, −2.01430570630827202000356084721, −1.00253527581590629456738759737, 0, 1.00253527581590629456738759737, 2.01430570630827202000356084721, 2.38817897782664356706357578105, 3.20467365999786331932186000044, 3.71784894557919390235888777459, 4.33941267588924836802297779070, 4.87696576807597648427388114522, 5.11339976412187934944202631996, 5.57099709901224003127203453141, 6.43277175562229230913123730651, 6.53502042577489063110653387067, 6.88886624979378547960752879584, 7.43041297806588240718290428045, 7.935322747816453095042923623228

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