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Properties

Label 4-311904-1.1-c1e2-0-29
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 + 8-s + 9-s − 12-s − 4·14-s + 16-s + 18-s − 4·19-s + 4·21-s − 24-s + 6·25-s − 27-s − 4·28-s − 4·29-s + 32-s + 36-s − 4·38-s + 16·41-s + 4·42-s − 48-s + 2·49-s + 6·50-s − 8·53-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.872·21-s − 0.204·24-s + 6/5·25-s − 0.192·27-s − 0.755·28-s − 0.742·29-s + 0.176·32-s + 1/6·36-s − 0.648·38-s + 2.49·41-s + 0.617·42-s − 0.144·48-s + 2/7·49-s + 0.848·50-s − 1.09·53-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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.e_o
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.e_ac
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.37.a_bu
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.41.aq_fm
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.i_di
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.m_fu
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
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$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ae_fm
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.u_iw
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.83.a_cc
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.e_bm
97$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.97.a_o
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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

−8.722004226747554232785503447987, −7.84137137987603428971278683803, −7.61290840027815430565926852347, −6.88258152399789669587522888774, −6.64130027425973879519373000253, −6.06803346755742853076135280648, −5.90551592739753488456858322419, −5.21870066662744913676894847088, −4.53114633194593558287948869469, −4.22760177787653697393370900481, −3.48448683380402607340751264018, −2.97579797594057402313735117934, −2.39184485111621331058665785058, −1.31937024815301995985138819457, 0, 1.31937024815301995985138819457, 2.39184485111621331058665785058, 2.97579797594057402313735117934, 3.48448683380402607340751264018, 4.22760177787653697393370900481, 4.53114633194593558287948869469, 5.21870066662744913676894847088, 5.90551592739753488456858322419, 6.06803346755742853076135280648, 6.64130027425973879519373000253, 6.88258152399789669587522888774, 7.61290840027815430565926852347, 7.84137137987603428971278683803, 8.722004226747554232785503447987

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