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Properties

Label 4-396508-1.1-c1e2-0-4
Degree $4$
Conductor $396508$
Sign $-1$
Analytic cond. $25.2816$
Root an. cond. $2.24234$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 7-s + 4·8-s − 6·9-s + 2·14-s + 5·16-s − 12·18-s − 6·25-s + 3·28-s − 12·29-s + 6·32-s − 18·36-s − 12·37-s − 24·43-s + 49-s − 12·50-s − 4·53-s + 4·56-s − 24·58-s − 6·63-s + 7·64-s + 24·67-s − 24·72-s − 24·74-s − 16·79-s + 27·81-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.377·7-s + 1.41·8-s − 2·9-s + 0.534·14-s + 5/4·16-s − 2.82·18-s − 6/5·25-s + 0.566·28-s − 2.22·29-s + 1.06·32-s − 3·36-s − 1.97·37-s − 3.65·43-s + 1/7·49-s − 1.69·50-s − 0.549·53-s + 0.534·56-s − 3.15·58-s − 0.755·63-s + 7/8·64-s + 2.93·67-s − 2.82·72-s − 2.78·74-s − 1.80·79-s + 3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396508 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396508 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396508\)    =    \(2^{2} \cdot 7^{3} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(25.2816\)
Root analytic conductor: \(2.24234\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 396508,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( 1 - T \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−8.224383160473151438327737126984, −8.141215318250912713100120450381, −7.31408639973055649098152315553, −6.94219495068550059915873452189, −6.42857794006623898982743840849, −5.70286626752820956259075840901, −5.68518485773994087792082977671, −5.11332775341498038219899119581, −4.74254867216183849876291306196, −3.82658470314743989463200974897, −3.40674497567057131140915818833, −3.15634367282858574822998301134, −2.04650857588819893440526220286, −1.91012210733034651779284644815, 0, 1.91012210733034651779284644815, 2.04650857588819893440526220286, 3.15634367282858574822998301134, 3.40674497567057131140915818833, 3.82658470314743989463200974897, 4.74254867216183849876291306196, 5.11332775341498038219899119581, 5.68518485773994087792082977671, 5.70286626752820956259075840901, 6.42857794006623898982743840849, 6.94219495068550059915873452189, 7.31408639973055649098152315553, 8.141215318250912713100120450381, 8.224383160473151438327737126984

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