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Properties

Label 4-961311-1.1-c1e2-0-3
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 + 4-s − 2·9-s − 11-s + 12-s + 3·13-s − 3·16-s + 8·17-s − 2·19-s − 25-s − 2·27-s − 13·31-s − 33-s − 2·36-s − 7·37-s + 3·39-s − 44-s − 3·47-s − 3·48-s − 5·49-s + 8·51-s + 3·52-s − 2·57-s − 7·64-s − 12·67-s + 8·68-s + 7·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s − 2/3·9-s − 0.301·11-s + 0.288·12-s + 0.832·13-s − 3/4·16-s + 1.94·17-s − 0.458·19-s − 1/5·25-s − 0.384·27-s − 2.33·31-s − 0.174·33-s − 1/3·36-s − 1.15·37-s + 0.480·39-s − 0.150·44-s − 0.437·47-s − 0.433·48-s − 5/7·49-s + 1.12·51-s + 0.416·52-s − 0.264·57-s − 7/8·64-s − 1.46·67-s + 0.970·68-s + 0.830·71-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$ \( ( 1 - T )( 1 + p T^{2} ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
157$C_2$ \( 1 + 11 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.11.b_w
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.c_bm
23$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.23.a_h
29$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.29.a_s
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.n_dy
37$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.h_di
41$C_2^2$ \( 1 - 51 T^{2} + p^{2} T^{4} \) 2.41.a_abz
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.43.a_aba
47$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.d_y
53$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.53.a_ae
59$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.59.a_abc
61$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \) 2.61.a_cx
67$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.67.m_gf
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.71.ah_fy
73$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.73.a_ay
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.79.a_abg
83$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.83.a_dq
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.ae_n
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.a_ack
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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

−7.934336586949930484322195014050, −7.50920687241841829085146875556, −7.21915202592452182321080115831, −6.50802296315077976390834735159, −6.23493114521968720384039862794, −5.66465487624548801166136700141, −5.27171458328237870117399242463, −4.88229575341483451492952196532, −3.94432508872565475591945924647, −3.59891442855010729414397492585, −3.18548235530684756677134070279, −2.59682177373666681893533850423, −1.92742500750037864745359695404, −1.36058927841735672452870773565, 0, 1.36058927841735672452870773565, 1.92742500750037864745359695404, 2.59682177373666681893533850423, 3.18548235530684756677134070279, 3.59891442855010729414397492585, 3.94432508872565475591945924647, 4.88229575341483451492952196532, 5.27171458328237870117399242463, 5.66465487624548801166136700141, 6.23493114521968720384039862794, 6.50802296315077976390834735159, 7.21915202592452182321080115831, 7.50920687241841829085146875556, 7.934336586949930484322195014050

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