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Properties

Label 4-311904-1.1-c1e2-0-13
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 − 3-s − 4-s − 6-s − 4·7-s − 3·8-s + 9-s + 12-s − 4·14-s − 16-s + 18-s + 8·19-s + 4·21-s + 3·24-s − 6·25-s − 27-s + 4·28-s + 5·32-s − 36-s + 8·38-s + 12·41-s + 4·42-s + 48-s + 2·49-s − 6·50-s + 8·53-s − 54-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.06·14-s − 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.872·21-s + 0.612·24-s − 6/5·25-s − 0.192·27-s + 0.755·28-s + 0.883·32-s − 1/6·36-s + 1.29·38-s + 1.87·41-s + 0.617·42-s + 0.144·48-s + 2/7·49-s − 0.848·50-s + 1.09·53-s − 0.136·54-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$C_2$ \( 1 - T + p T^{2} \)
3$C_1$ \( 1 + T \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.e_o
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.am_dy
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.47.a_ak
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.ai_w
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.q_gk
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.67.a_g
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.97.a_ck
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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

−8.736889791335082535304097324446, −7.941285762639805320237806266424, −7.58005456826407510059411641598, −7.11368435429897210182678491780, −6.43130195960201556922275445607, −6.07614964355063471924975790959, −5.73936479122931926421601756851, −5.25314988953772422944779442889, −4.64374511669624399068122590216, −4.10248946608386841404593084716, −3.48916350919933185414807438401, −3.15805717781871921269986140647, −2.42355400058332832135232946533, −1.09249834519731059921741391320, 0, 1.09249834519731059921741391320, 2.42355400058332832135232946533, 3.15805717781871921269986140647, 3.48916350919933185414807438401, 4.10248946608386841404593084716, 4.64374511669624399068122590216, 5.25314988953772422944779442889, 5.73936479122931926421601756851, 6.07614964355063471924975790959, 6.43130195960201556922275445607, 7.11368435429897210182678491780, 7.58005456826407510059411641598, 7.941285762639805320237806266424, 8.736889791335082535304097324446

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