Properties

Label 2-2738-1.1-c1-0-80
Degree $2$
Conductor $2738$
Sign $1$
Analytic cond. $21.8630$
Root an. cond. $4.67579$
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 + 3.30·3-s + 4-s + 2.30·5-s + 3.30·6-s − 2.60·7-s + 8-s + 7.90·9-s + 2.30·10-s − 2.30·11-s + 3.30·12-s − 1.30·13-s − 2.60·14-s + 7.60·15-s + 16-s + 6·17-s + 7.90·18-s − 2·19-s + 2.30·20-s − 8.60·21-s − 2.30·22-s − 3.90·23-s + 3.30·24-s + 0.302·25-s − 1.30·26-s + 16.2·27-s − 2.60·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.90·3-s + 0.5·4-s + 1.02·5-s + 1.34·6-s − 0.984·7-s + 0.353·8-s + 2.63·9-s + 0.728·10-s − 0.694·11-s + 0.953·12-s − 0.361·13-s − 0.696·14-s + 1.96·15-s + 0.250·16-s + 1.45·17-s + 1.86·18-s − 0.458·19-s + 0.514·20-s − 1.87·21-s − 0.490·22-s − 0.814·23-s + 0.674·24-s + 0.0605·25-s − 0.255·26-s + 3.11·27-s − 0.492·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2738\)    =    \(2 \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(21.8630\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2738} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2738,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.216100945\)
\(L(\frac12)\) \(\approx\) \(6.216100945\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
37 \( 1 \)
good3 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + 1.30T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.90T + 23T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 - 0.302T + 31T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 + 0.605T + 43T^{2} \)
47 \( 1 - 4.60T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + 7.51T + 61T^{2} \)
67 \( 1 + 3.51T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + 9.11T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−8.944968886000003926650433398828, −7.86056068938729605223363819641, −7.56462152729171433762411344891, −6.46326724060365295881527750344, −5.83414971611436273708895190435, −4.72031344853855943081692986728, −3.80025024914192379694990640252, −2.96548774308850045829590794528, −2.52714308561159724942380146071, −1.54305188307646409208210980611, 1.54305188307646409208210980611, 2.52714308561159724942380146071, 2.96548774308850045829590794528, 3.80025024914192379694990640252, 4.72031344853855943081692986728, 5.83414971611436273708895190435, 6.46326724060365295881527750344, 7.56462152729171433762411344891, 7.86056068938729605223363819641, 8.944968886000003926650433398828

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