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Properties

Label 4-311904-1.1-c1e2-0-35
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 + 8-s + 9-s − 12-s + 16-s + 18-s − 4·19-s − 24-s − 6·25-s − 27-s − 4·29-s + 32-s + 36-s − 4·38-s − 12·41-s − 8·43-s − 48-s − 14·49-s − 6·50-s + 20·53-s − 54-s + 4·57-s − 4·58-s + 16·59-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.204·24-s − 6/5·25-s − 0.192·27-s − 0.742·29-s + 0.176·32-s + 1/6·36-s − 0.648·38-s − 1.87·41-s − 1.21·43-s − 0.144·48-s − 2·49-s − 0.848·50-s + 2.74·53-s − 0.136·54-s + 0.529·57-s − 0.525·58-s + 2.08·59-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$C_1$ \( 1 + T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.37.a_as
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.53.au_hi
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.aq_gk
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.m_fm
67$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.67.a_aec
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \) 2.79.a_aes
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.e_bm
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

−8.565624415378647973335060348448, −8.093027696258890903844982944309, −7.48141419590004485294178768762, −7.03413405283409759862266438597, −6.65821052891035658426720424305, −6.06460192375792306110998505550, −5.74438286891327077413779181910, −5.13047739965993028491477828844, −4.76434682235528064136085705255, −4.02423816127241355215978975673, −3.71653948518943110660099077738, −2.96384737810928641099071647807, −2.10551341230024945663696972418, −1.54017298869936497866543931819, 0, 1.54017298869936497866543931819, 2.10551341230024945663696972418, 2.96384737810928641099071647807, 3.71653948518943110660099077738, 4.02423816127241355215978975673, 4.76434682235528064136085705255, 5.13047739965993028491477828844, 5.74438286891327077413779181910, 6.06460192375792306110998505550, 6.65821052891035658426720424305, 7.03413405283409759862266438597, 7.48141419590004485294178768762, 8.093027696258890903844982944309, 8.565624415378647973335060348448

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