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Properties

Label 4-311904-1.1-c1e2-0-41
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  + 2·2-s + 3-s + 2·4-s + 2·6-s − 4·7-s + 9-s + 2·12-s − 8·14-s − 4·16-s + 2·18-s − 4·19-s − 4·21-s − 3·25-s + 27-s − 8·28-s + 6·29-s − 8·32-s + 2·36-s − 8·38-s − 12·41-s − 8·42-s − 4·48-s − 49-s − 6·50-s − 8·53-s + 2·54-s − 4·57-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 0.577·12-s − 2.13·14-s − 16-s + 0.471·18-s − 0.917·19-s − 0.872·21-s − 3/5·25-s + 0.192·27-s − 1.51·28-s + 1.11·29-s − 1.41·32-s + 1/3·36-s − 1.29·38-s − 1.87·41-s − 1.23·42-s − 0.577·48-s − 1/7·49-s − 0.848·50-s − 1.09·53-s + 0.272·54-s − 0.529·57-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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.e_r
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.17.a_ab
23$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.23.a_aw
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.47.a_bp
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.i_cg
59$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.o_fm
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.61.ao_gp
67$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.67.a_ek
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.g_cs
73$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.c_eh
79$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.79.a_de
83$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.83.a_adi
89$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.ae_fq
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−8.526155715989749768823275864207, −8.162495825002542528669348073669, −7.52506162576454829865147104617, −6.87009989324006311865291589130, −6.44439137125381993238647794818, −6.37636240061940899313585652404, −5.69227187413300521312327295700, −5.04796383756659429699174417104, −4.58781969936242355146789276932, −4.01184686911776553242809910421, −3.46064567725120525163326359421, −3.10037126327037775382032879092, −2.54996623653434474549483400456, −1.73017569030778596946988540021, 0, 1.73017569030778596946988540021, 2.54996623653434474549483400456, 3.10037126327037775382032879092, 3.46064567725120525163326359421, 4.01184686911776553242809910421, 4.58781969936242355146789276932, 5.04796383756659429699174417104, 5.69227187413300521312327295700, 6.37636240061940899313585652404, 6.44439137125381993238647794818, 6.87009989324006311865291589130, 7.52506162576454829865147104617, 8.162495825002542528669348073669, 8.526155715989749768823275864207

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