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Properties

Label 4-57967-1.1-c1e2-0-1
Degree $4$
Conductor $57967$
Sign $1$
Analytic cond. $3.69602$
Root an. cond. $1.38654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 7-s − 8·8-s − 6·9-s − 12·11-s + 4·14-s − 4·16-s + 24·18-s + 48·22-s + 6·23-s − 25-s − 8·28-s − 10·29-s + 32·32-s − 48·36-s − 8·37-s − 2·43-s − 96·44-s − 24·46-s + 49-s + 4·50-s − 18·53-s + 8·56-s + 40·58-s + 6·63-s − 64·64-s + ⋯
L(s)  = 1  − 2.82·2-s + 4·4-s − 0.377·7-s − 2.82·8-s − 2·9-s − 3.61·11-s + 1.06·14-s − 16-s + 5.65·18-s + 10.2·22-s + 1.25·23-s − 1/5·25-s − 1.51·28-s − 1.85·29-s + 5.65·32-s − 8·36-s − 1.31·37-s − 0.304·43-s − 14.4·44-s − 3.53·46-s + 1/7·49-s + 0.565·50-s − 2.47·53-s + 1.06·56-s + 5.25·58-s + 0.755·63-s − 8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57967 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57967 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57967\)    =    \(7^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3.69602\)
Root analytic conductor: \(1.38654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 57967,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad7$C_1$ \( 1 + T \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \) 2.2.e_i
3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.67.m_go
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.a_ax
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.79.ag_gl
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.a_fp
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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

−9.232409444628192264737671791785, −8.998346772410808154035420365472, −8.568760068955747319631158439108, −7.88305646625584781075065151406, −7.85318737682082614131812951371, −7.43563929744309328324545511818, −6.67298853468060181727432085852, −5.78821104763479917640596143353, −5.27883313770925671699270459709, −4.82450768688560173606084932648, −3.11744276397134147631028514628, −2.72528535271701611265854270606, −1.93185778189567456463072050241, 0, 0, 1.93185778189567456463072050241, 2.72528535271701611265854270606, 3.11744276397134147631028514628, 4.82450768688560173606084932648, 5.27883313770925671699270459709, 5.78821104763479917640596143353, 6.67298853468060181727432085852, 7.43563929744309328324545511818, 7.85318737682082614131812951371, 7.88305646625584781075065151406, 8.568760068955747319631158439108, 8.998346772410808154035420365472, 9.232409444628192264737671791785

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