Properties

Label 4-980000-1.1-c1e2-0-11
Degree $4$
Conductor $980000$
Sign $1$
Analytic cond. $62.4856$
Root an. cond. $2.81154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 5·9-s − 4·13-s + 16-s − 6·17-s + 5·18-s + 4·26-s − 12·29-s − 32-s + 6·34-s − 5·36-s − 16·37-s − 18·41-s + 49-s − 4·52-s + 24·53-s + 12·58-s − 20·61-s + 64-s − 6·68-s + 5·72-s − 10·73-s + 16·74-s + 16·81-s + 18·82-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 5/3·9-s − 1.10·13-s + 1/4·16-s − 1.45·17-s + 1.17·18-s + 0.784·26-s − 2.22·29-s − 0.176·32-s + 1.02·34-s − 5/6·36-s − 2.63·37-s − 2.81·41-s + 1/7·49-s − 0.554·52-s + 3.29·53-s + 1.57·58-s − 2.56·61-s + 1/8·64-s − 0.727·68-s + 0.589·72-s − 1.17·73-s + 1.85·74-s + 16/9·81-s + 1.98·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(980000\)    =    \(2^{5} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(62.4856\)
Root analytic conductor: \(2.81154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{980000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 980000,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( 1 + T \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.55390913417171576921785305266, −7.29532124265233143164283708491, −7.02864082573576697328686349834, −6.40152272675796392235796567725, −5.97109864819728246690045551416, −5.30999251676143514149579674594, −5.26770803610804114861303773339, −4.55568629905960458761269368363, −3.76060769659553506276694871497, −3.34803193788577441588970996410, −2.72399279449132066058144547784, −2.12269688406492296074462119899, −1.71004502018900947247065247921, 0, 0, 1.71004502018900947247065247921, 2.12269688406492296074462119899, 2.72399279449132066058144547784, 3.34803193788577441588970996410, 3.76060769659553506276694871497, 4.55568629905960458761269368363, 5.26770803610804114861303773339, 5.30999251676143514149579674594, 5.97109864819728246690045551416, 6.40152272675796392235796567725, 7.02864082573576697328686349834, 7.29532124265233143164283708491, 7.55390913417171576921785305266

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