Properties

Label 2-7500-1.1-c1-0-29
Degree $2$
Conductor $7500$
Sign $1$
Analytic cond. $59.8878$
Root an. cond. $7.73872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 1.04·7-s + 9-s + 6.28·11-s − 1.00·13-s − 4.69·17-s − 5.97·19-s + 1.04·21-s + 8.05·23-s + 27-s + 6.91·29-s − 9.52·31-s + 6.28·33-s + 7.69·37-s − 1.00·39-s + 1.56·41-s + 9.94·43-s − 4.84·47-s − 5.90·49-s − 4.69·51-s + 2.82·53-s − 5.97·57-s + 4.08·59-s + 3.16·61-s + 1.04·63-s + 3.17·67-s + 8.05·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.395·7-s + 0.333·9-s + 1.89·11-s − 0.279·13-s − 1.13·17-s − 1.37·19-s + 0.228·21-s + 1.67·23-s + 0.192·27-s + 1.28·29-s − 1.71·31-s + 1.09·33-s + 1.26·37-s − 0.161·39-s + 0.245·41-s + 1.51·43-s − 0.706·47-s − 0.843·49-s − 0.656·51-s + 0.388·53-s − 0.791·57-s + 0.531·59-s + 0.404·61-s + 0.131·63-s + 0.387·67-s + 0.969·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7500\)    =    \(2^{2} \cdot 3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(59.8878\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7500,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.057864064\)
\(L(\frac12)\) \(\approx\) \(3.057864064\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
good7 \( 1 - 1.04T + 7T^{2} \)
11 \( 1 - 6.28T + 11T^{2} \)
13 \( 1 + 1.00T + 13T^{2} \)
17 \( 1 + 4.69T + 17T^{2} \)
19 \( 1 + 5.97T + 19T^{2} \)
23 \( 1 - 8.05T + 23T^{2} \)
29 \( 1 - 6.91T + 29T^{2} \)
31 \( 1 + 9.52T + 31T^{2} \)
37 \( 1 - 7.69T + 37T^{2} \)
41 \( 1 - 1.56T + 41T^{2} \)
43 \( 1 - 9.94T + 43T^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 - 2.82T + 53T^{2} \)
59 \( 1 - 4.08T + 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 - 6.50T + 71T^{2} \)
73 \( 1 + 0.367T + 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 2.50T + 89T^{2} \)
97 \( 1 - 0.182T + 97T^{2} \)
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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

−8.020935698240477378493982967995, −6.95529146790083196940238400797, −6.76772781929826329272207989602, −5.94415650067527949087107177273, −4.80457923655169771997325252861, −4.30079223076247674874992907641, −3.64687266052152763531932865348, −2.60905392912489700875386288981, −1.85300344574785426663291488849, −0.888981798532948323438824504677, 0.888981798532948323438824504677, 1.85300344574785426663291488849, 2.60905392912489700875386288981, 3.64687266052152763531932865348, 4.30079223076247674874992907641, 4.80457923655169771997325252861, 5.94415650067527949087107177273, 6.76772781929826329272207989602, 6.95529146790083196940238400797, 8.020935698240477378493982967995

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