Properties

Label 4-980e2-1.1-c1e2-0-14
Degree $4$
Conductor $960400$
Sign $-1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 2·5-s − 5·9-s − 10·13-s + 4·16-s − 6·17-s − 4·20-s + 3·25-s + 6·29-s + 10·36-s + 4·37-s + 24·41-s − 10·45-s + 20·52-s + 24·53-s − 16·61-s − 8·64-s − 20·65-s + 12·68-s − 4·73-s + 8·80-s + 16·81-s − 12·85-s + 24·89-s + 2·97-s − 6·100-s − 12·101-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s − 5/3·9-s − 2.77·13-s + 16-s − 1.45·17-s − 0.894·20-s + 3/5·25-s + 1.11·29-s + 5/3·36-s + 0.657·37-s + 3.74·41-s − 1.49·45-s + 2.77·52-s + 3.29·53-s − 2.04·61-s − 64-s − 2.48·65-s + 1.45·68-s − 0.468·73-s + 0.894·80-s + 16/9·81-s − 1.30·85-s + 2.54·89-s + 0.203·97-s − 3/5·100-s − 1.19·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{960400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ -1)\)

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

−7.80156257853413862136981277007, −7.59679103544400835112633947355, −7.18840908028866321022100981317, −6.30078072165756181986797640838, −6.23576189288857341841080164821, −5.51580216499301883688612135353, −5.30121167828360660438204787375, −4.66901616031576415432379359350, −4.48390865811726709341858080699, −3.81894678634232741164227871759, −2.72282955438852544862637314814, −2.61198254943438177491615875407, −2.28641578958231941287903879491, −0.863841609060871512635709852891, 0, 0.863841609060871512635709852891, 2.28641578958231941287903879491, 2.61198254943438177491615875407, 2.72282955438852544862637314814, 3.81894678634232741164227871759, 4.48390865811726709341858080699, 4.66901616031576415432379359350, 5.30121167828360660438204787375, 5.51580216499301883688612135353, 6.23576189288857341841080164821, 6.30078072165756181986797640838, 7.18840908028866321022100981317, 7.59679103544400835112633947355, 7.80156257853413862136981277007

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