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Properties

Label 4-311904-1.1-c1e2-0-21
Degree $4$
Conductor $311904$
Sign $1$
Analytic cond. $19.8872$
Root an. cond. $2.11175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8·7-s + 8-s + 9-s − 12-s + 8·14-s + 16-s + 18-s − 8·21-s − 24-s + 2·25-s − 27-s + 8·28-s + 32-s + 36-s + 8·41-s − 8·42-s − 16·43-s − 48-s + 34·49-s + 2·50-s − 8·53-s − 54-s + 8·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 3.02·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 2.13·14-s + 1/4·16-s + 0.235·18-s − 1.74·21-s − 0.204·24-s + 2/5·25-s − 0.192·27-s + 1.51·28-s + 0.176·32-s + 1/6·36-s + 1.24·41-s − 1.23·42-s − 2.43·43-s − 0.144·48-s + 34/7·49-s + 0.282·50-s − 1.09·53-s − 0.136·54-s + 1.06·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(311904\)    =    \(2^{5} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(19.8872\)
Root analytic conductor: \(2.11175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 311904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.580024093\)
\(L(\frac12)\) \(\approx\) \(3.580024093\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.q_fe
47$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.47.a_as
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.i_w
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ae_eg
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.ai_o
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ae_di
79$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.79.a_aco
83$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.83.a_ew
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.97.a_as
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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

−8.507715511114075286894175141063, −8.200344744121783587437076252308, −7.949067736289992134363590305738, −7.44641456275668954082085700814, −6.86064241100651556020553811470, −6.45718103498948306679141090642, −5.67271794243965818868344665930, −5.24966906946511307201231762954, −5.00655429413559990230305718790, −4.47546356248783832677725853301, −4.14274337006079054899353257789, −3.28949574697702606619855994393, −2.34186027881732176539224266375, −1.73866707395134719760996459126, −1.18065767260149891896117752833, 1.18065767260149891896117752833, 1.73866707395134719760996459126, 2.34186027881732176539224266375, 3.28949574697702606619855994393, 4.14274337006079054899353257789, 4.47546356248783832677725853301, 5.00655429413559990230305718790, 5.24966906946511307201231762954, 5.67271794243965818868344665930, 6.45718103498948306679141090642, 6.86064241100651556020553811470, 7.44641456275668954082085700814, 7.949067736289992134363590305738, 8.200344744121783587437076252308, 8.507715511114075286894175141063

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