Properties

Label 4-350e2-1.1-c1e2-0-17
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  − 4-s + 5·9-s + 6·11-s + 16-s + 14·19-s + 12·29-s − 8·31-s − 5·36-s − 18·41-s − 6·44-s − 49-s − 24·59-s − 20·61-s − 64-s + 12·71-s − 14·76-s − 28·79-s + 16·81-s + 30·89-s + 30·99-s − 28·109-s − 12·116-s + 5·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1.80·11-s + 1/4·16-s + 3.21·19-s + 2.22·29-s − 1.43·31-s − 5/6·36-s − 2.81·41-s − 0.904·44-s − 1/7·49-s − 3.12·59-s − 2.56·61-s − 1/8·64-s + 1.42·71-s − 1.60·76-s − 3.15·79-s + 16/9·81-s + 3.17·89-s + 3.01·99-s − 2.68·109-s − 1.11·116-s + 5/11·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(1.977753131\)
\(L(\frac12)\) \(\approx\) \(1.977753131\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−11.92881269295989137345621351552, −11.43542080794687841361161320888, −10.47171892529940308758171697404, −10.47158512922332734982283242374, −9.579009428886797063318045648029, −9.568685522922711961902715967641, −9.137854964217652677403368906993, −8.585608861462356437225398783181, −7.66578944834140103048419730306, −7.64782974353942160188456327130, −6.81224159074683869205437025071, −6.67972868613410665024264603752, −5.92173721221202844095837243996, −5.12974229193952936245223353764, −4.75532718895897824441470639549, −4.22140969165292187868312288693, −3.38090030408863777519563102748, −3.22476268795815471130630124211, −1.39931523367926616697580766515, −1.38643737698553398363216844504, 1.38643737698553398363216844504, 1.39931523367926616697580766515, 3.22476268795815471130630124211, 3.38090030408863777519563102748, 4.22140969165292187868312288693, 4.75532718895897824441470639549, 5.12974229193952936245223353764, 5.92173721221202844095837243996, 6.67972868613410665024264603752, 6.81224159074683869205437025071, 7.64782974353942160188456327130, 7.66578944834140103048419730306, 8.585608861462356437225398783181, 9.137854964217652677403368906993, 9.568685522922711961902715967641, 9.579009428886797063318045648029, 10.47158512922332734982283242374, 10.47171892529940308758171697404, 11.43542080794687841361161320888, 11.92881269295989137345621351552

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