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Properties

Label 4-676000-1.1-c1e2-0-4
Degree $4$
Conductor $676000$
Sign $1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 4·9-s − 10-s − 4·13-s + 16-s + 6·17-s − 4·18-s − 20-s + 25-s − 4·26-s + 6·29-s + 32-s + 6·34-s − 4·36-s − 4·37-s − 40-s − 12·41-s + 4·45-s + 14·49-s + 50-s − 4·52-s + 12·53-s + 6·58-s + 2·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 4/3·9-s − 0.316·10-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.942·18-s − 0.223·20-s + 1/5·25-s − 0.784·26-s + 1.11·29-s + 0.176·32-s + 1.02·34-s − 2/3·36-s − 0.657·37-s − 0.158·40-s − 1.87·41-s + 0.596·45-s + 2·49-s + 0.141·50-s − 0.554·52-s + 1.64·53-s + 0.787·58-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.144941969\)
\(L(\frac12)\) \(\approx\) \(2.144941969\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ag_bi
19$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.19.a_aq
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.a_ba
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.59.a_dc
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.ac_bq
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.71.a_aec
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.s_jq
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.359950296947014877365981450840, −7.87619022469839912600888296910, −7.34325933404267656403721502413, −7.01646301068502592735819979306, −6.58436162754623358856484204576, −5.86429773067408501297565048578, −5.49174562570121925439242783376, −5.24698777276131116365787022300, −4.66694596446837518403017463427, −4.03883036206393027786035887662, −3.53285416981378619719451150072, −2.88060077707522469887102498228, −2.68029657148873588697477586364, −1.75752814003110947064173232465, −0.63922013635487582447240227762, 0.63922013635487582447240227762, 1.75752814003110947064173232465, 2.68029657148873588697477586364, 2.88060077707522469887102498228, 3.53285416981378619719451150072, 4.03883036206393027786035887662, 4.66694596446837518403017463427, 5.24698777276131116365787022300, 5.49174562570121925439242783376, 5.86429773067408501297565048578, 6.58436162754623358856484204576, 7.01646301068502592735819979306, 7.34325933404267656403721502413, 7.87619022469839912600888296910, 8.359950296947014877365981450840

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