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Properties

Label 4-311904-1.1-c1e2-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 6·5-s − 8-s − 3·9-s + 6·10-s + 16-s + 3·17-s + 3·18-s − 6·19-s − 6·20-s + 18·25-s + 3·31-s − 32-s − 3·34-s − 3·36-s + 6·38-s + 6·40-s + 18·45-s + 4·49-s − 18·50-s + 3·59-s + 3·61-s − 3·62-s + 64-s + 6·67-s + 3·68-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 2.68·5-s − 0.353·8-s − 9-s + 1.89·10-s + 1/4·16-s + 0.727·17-s + 0.707·18-s − 1.37·19-s − 1.34·20-s + 18/5·25-s + 0.538·31-s − 0.176·32-s − 0.514·34-s − 1/2·36-s + 0.973·38-s + 0.948·40-s + 2.68·45-s + 4/7·49-s − 2.54·50-s + 0.390·59-s + 0.384·61-s − 0.381·62-s + 1/8·64-s + 0.733·67-s + 0.363·68-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: \(1\)
Selberg data: \((4,\ 311904,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_2$ \( 1 + p T^{2} \)
19$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.g_s
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.13.a_ai
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ad_y
23$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.23.a_ap
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.29.a_n
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.ad_bs
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.37.a_au
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.41.a_ai
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.ad_k
61$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ad_es
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.67.ag_fe
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.ag_cs
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ao_ek
79$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.ad_aw
83$C_2^2$ \( 1 - 36 T^{2} + p^{2} T^{4} \) 2.83.a_abk
89$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.89.a_abm
97$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.a_adr
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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

−8.342730333480456406579974123015, −8.139499671608587532131930780785, −7.890226509545634392765080740470, −7.38365419586850268326170785764, −6.72180661032823708492047279344, −6.57252416669905646537628050663, −5.67114705454295229662265039811, −5.20162348293338895653181840361, −4.40662126530608505167961906401, −3.99953713969694421860649453213, −3.53504961735030896761622599530, −2.96010347213561316878632444656, −2.23109421820828999676631183520, −0.824797561920330696264374355881, 0, 0.824797561920330696264374355881, 2.23109421820828999676631183520, 2.96010347213561316878632444656, 3.53504961735030896761622599530, 3.99953713969694421860649453213, 4.40662126530608505167961906401, 5.20162348293338895653181840361, 5.67114705454295229662265039811, 6.57252416669905646537628050663, 6.72180661032823708492047279344, 7.38365419586850268326170785764, 7.890226509545634392765080740470, 8.139499671608587532131930780785, 8.342730333480456406579974123015

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