Properties

Label 2-1502-1.1-c1-0-36
Degree $2$
Conductor $1502$
Sign $1$
Analytic cond. $11.9935$
Root an. cond. $3.46316$
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  + 2-s + 2.57·3-s + 4-s − 0.396·5-s + 2.57·6-s − 0.271·7-s + 8-s + 3.62·9-s − 0.396·10-s − 3.20·11-s + 2.57·12-s + 4.71·13-s − 0.271·14-s − 1.02·15-s + 16-s + 6.19·17-s + 3.62·18-s + 1.43·19-s − 0.396·20-s − 0.698·21-s − 3.20·22-s − 1.57·23-s + 2.57·24-s − 4.84·25-s + 4.71·26-s + 1.60·27-s − 0.271·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.48·3-s + 0.5·4-s − 0.177·5-s + 1.05·6-s − 0.102·7-s + 0.353·8-s + 1.20·9-s − 0.125·10-s − 0.966·11-s + 0.742·12-s + 1.30·13-s − 0.0725·14-s − 0.263·15-s + 0.250·16-s + 1.50·17-s + 0.853·18-s + 0.328·19-s − 0.0886·20-s − 0.152·21-s − 0.683·22-s − 0.328·23-s + 0.525·24-s − 0.968·25-s + 0.923·26-s + 0.308·27-s − 0.0512·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1502\)    =    \(2 \cdot 751\)
Sign: $1$
Analytic conductor: \(11.9935\)
Root analytic conductor: \(3.46316\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1502,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.322104932\)
\(L(\frac12)\) \(\approx\) \(4.322104932\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
751 \( 1 + T \)
good3 \( 1 - 2.57T + 3T^{2} \)
5 \( 1 + 0.396T + 5T^{2} \)
7 \( 1 + 0.271T + 7T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 - 6.19T + 17T^{2} \)
19 \( 1 - 1.43T + 19T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 - 5.70T + 29T^{2} \)
31 \( 1 - 3.42T + 31T^{2} \)
37 \( 1 + 9.40T + 37T^{2} \)
41 \( 1 + 0.165T + 41T^{2} \)
43 \( 1 - 7.51T + 43T^{2} \)
47 \( 1 - 2.84T + 47T^{2} \)
53 \( 1 + 6.33T + 53T^{2} \)
59 \( 1 + 2.89T + 59T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 - 8.50T + 67T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 + 1.26T + 73T^{2} \)
79 \( 1 - 0.454T + 79T^{2} \)
83 \( 1 + 7.76T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 9.38T + 97T^{2} \)
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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

−9.482934313399625763351903367199, −8.346595033580540002568001842467, −8.060687965391981573250696570368, −7.25835475974545773790994478142, −6.12738949235361523028752377644, −5.29421457150878055318472130792, −4.09679229632387760879100928336, −3.36354984580986178902496773710, −2.73694193582784129606419679019, −1.49228412830844895693333990469, 1.49228412830844895693333990469, 2.73694193582784129606419679019, 3.36354984580986178902496773710, 4.09679229632387760879100928336, 5.29421457150878055318472130792, 6.12738949235361523028752377644, 7.25835475974545773790994478142, 8.060687965391981573250696570368, 8.346595033580540002568001842467, 9.482934313399625763351903367199

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