Properties

Label 2-50e2-4.3-c0-0-1
Degree $2$
Conductor $2500$
Sign $1$
Analytic cond. $1.24766$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 1.61·13-s + 16-s − 0.618·17-s − 18-s − 1.61·26-s + 0.618·29-s − 32-s + 0.618·34-s + 36-s − 0.618·37-s − 1.61·41-s + 49-s + 1.61·52-s + 1.61·53-s − 0.618·58-s − 1.61·61-s + 64-s − 0.618·68-s − 72-s + 1.61·73-s + 0.618·74-s + 81-s + 1.61·82-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 1.61·13-s + 16-s − 0.618·17-s − 18-s − 1.61·26-s + 0.618·29-s − 32-s + 0.618·34-s + 36-s − 0.618·37-s − 1.61·41-s + 49-s + 1.61·52-s + 1.61·53-s − 0.618·58-s − 1.61·61-s + 64-s − 0.618·68-s − 72-s + 1.61·73-s + 0.618·74-s + 81-s + 1.61·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1251, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9023899696\)
\(L(\frac12)\) \(\approx\) \(0.9023899696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
good3 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 + 0.618T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 0.618T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.61T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + 0.618T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.944191339179056758140172353089, −8.522717546213237955688312177516, −7.67860420595458069790843664791, −6.82384297810429551325557449673, −6.37250204457914593014868612169, −5.36046846446162242401468695755, −4.15073267321779786561828287260, −3.31625537045838323918266477847, −2.04785826340634829589181124630, −1.12162084521037929633987422510, 1.12162084521037929633987422510, 2.04785826340634829589181124630, 3.31625537045838323918266477847, 4.15073267321779786561828287260, 5.36046846446162242401468695755, 6.37250204457914593014868612169, 6.82384297810429551325557449673, 7.67860420595458069790843664791, 8.522717546213237955688312177516, 8.944191339179056758140172353089

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