Properties

Label 4-350e2-1.1-c1e2-0-14
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $7.81070$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4·7-s − 8-s + 3·9-s − 3·11-s + 10·13-s + 4·14-s − 16-s − 2·17-s + 3·18-s + 5·19-s − 3·22-s − 7·23-s + 10·26-s − 8·29-s + 2·31-s − 2·34-s + 37-s + 5·38-s + 6·41-s − 4·43-s − 7·46-s − 7·47-s + 9·49-s + 9·53-s − 4·56-s − 8·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.51·7-s − 0.353·8-s + 9-s − 0.904·11-s + 2.77·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 1.14·19-s − 0.639·22-s − 1.45·23-s + 1.96·26-s − 1.48·29-s + 0.359·31-s − 0.342·34-s + 0.164·37-s + 0.811·38-s + 0.937·41-s − 0.609·43-s − 1.03·46-s − 1.02·47-s + 9/7·49-s + 1.23·53-s − 0.534·56-s − 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.81070\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{350} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.898939002\)
\(L(\frac12)\) \(\approx\) \(2.898939002\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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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

−11.66323269048498479280085956263, −11.43037689665071350163098355732, −10.68177613968475580988606415535, −10.67423866995848936287344453353, −9.990005599916454234639780513615, −9.388118800271014754235695322422, −8.792905082338304325384859022469, −8.432844752820524408629777323891, −7.897430831147115771947835116270, −7.60537317777427137050936828727, −6.95807417259234074157050225892, −6.18504823601605595008371513547, −5.77708077183459598292002559870, −5.37509321581881459979995080225, −4.67526715532093570447953523581, −4.07877084750982916932396334106, −3.84342406883718544027527528705, −2.96050640500358563979940233413, −1.84383325856557009391926571129, −1.31034123696361728243552015324, 1.31034123696361728243552015324, 1.84383325856557009391926571129, 2.96050640500358563979940233413, 3.84342406883718544027527528705, 4.07877084750982916932396334106, 4.67526715532093570447953523581, 5.37509321581881459979995080225, 5.77708077183459598292002559870, 6.18504823601605595008371513547, 6.95807417259234074157050225892, 7.60537317777427137050936828727, 7.897430831147115771947835116270, 8.432844752820524408629777323891, 8.792905082338304325384859022469, 9.388118800271014754235695322422, 9.990005599916454234639780513615, 10.67423866995848936287344453353, 10.68177613968475580988606415535, 11.43037689665071350163098355732, 11.66323269048498479280085956263

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