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Properties

Label 4-50128-1.1-c1e2-0-2
Degree $4$
Conductor $50128$
Sign $-1$
Analytic cond. $3.19620$
Root an. cond. $1.33708$
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 + 2·4-s + 5-s − 9-s − 2·10-s − 3·13-s − 4·16-s − 3·17-s + 2·18-s + 2·20-s − 7·25-s + 6·26-s − 2·29-s + 8·32-s + 6·34-s − 2·36-s + 6·37-s − 22·41-s − 45-s − 11·49-s + 14·50-s − 6·52-s + 8·53-s + 4·58-s + 2·61-s − 8·64-s − 3·65-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 1/3·9-s − 0.632·10-s − 0.832·13-s − 16-s − 0.727·17-s + 0.471·18-s + 0.447·20-s − 7/5·25-s + 1.17·26-s − 0.371·29-s + 1.41·32-s + 1.02·34-s − 1/3·36-s + 0.986·37-s − 3.43·41-s − 0.149·45-s − 1.57·49-s + 1.97·50-s − 0.832·52-s + 1.09·53-s + 0.525·58-s + 0.256·61-s − 64-s − 0.372·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50128\)    =    \(2^{4} \cdot 13 \cdot 241\)
Sign: $-1$
Analytic conductor: \(3.19620\)
Root analytic conductor: \(1.33708\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 50128,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
241$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.ab_i
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.11.a_af
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.c_cg
31$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.31.a_p
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ag_de
41$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.w_hu
43$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.43.a_au
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.47.a_bs
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.53.ai_dt
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.ac_cw
67$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.67.a_ca
71$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.71.a_abh
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.79.a_bh
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.83.a_cq
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.q_hl
97$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.b_hm
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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

−9.769346879936481424559325424075, −9.472620833418092111425924119800, −8.819546499052649986503950757488, −8.312450294669500562952032826528, −8.040657201409478460089658955648, −7.24297270847229282671883867452, −6.92263680349130291623559341905, −6.28592701230375960766148748810, −5.59387737857555542352140829553, −4.92794756493464535389927901551, −4.27623478416879547016175581876, −3.29328033381916729509335906904, −2.30120820507375229568855611704, −1.69171992717569139656582877115, 0, 1.69171992717569139656582877115, 2.30120820507375229568855611704, 3.29328033381916729509335906904, 4.27623478416879547016175581876, 4.92794756493464535389927901551, 5.59387737857555542352140829553, 6.28592701230375960766148748810, 6.92263680349130291623559341905, 7.24297270847229282671883867452, 8.040657201409478460089658955648, 8.312450294669500562952032826528, 8.819546499052649986503950757488, 9.472620833418092111425924119800, 9.769346879936481424559325424075

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