Properties

Label 2-2738-1.1-c1-0-39
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 − 1.87·3-s + 4-s + 4.26·5-s + 1.87·6-s + 3.61·7-s − 8-s + 0.532·9-s − 4.26·10-s + 2.11·11-s − 1.87·12-s + 2.88·13-s − 3.61·14-s − 8.01·15-s + 16-s − 0.0267·17-s − 0.532·18-s + 3.22·19-s + 4.26·20-s − 6.78·21-s − 2.11·22-s − 2.97·23-s + 1.87·24-s + 13.1·25-s − 2.88·26-s + 4.63·27-s + 3.61·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.08·3-s + 0.5·4-s + 1.90·5-s + 0.767·6-s + 1.36·7-s − 0.353·8-s + 0.177·9-s − 1.34·10-s + 0.638·11-s − 0.542·12-s + 0.799·13-s − 0.965·14-s − 2.06·15-s + 0.250·16-s − 0.00648·17-s − 0.125·18-s + 0.740·19-s + 0.953·20-s − 1.48·21-s − 0.451·22-s − 0.619·23-s + 0.383·24-s + 2.63·25-s − 0.565·26-s + 0.892·27-s + 0.682·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2738,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.740334842\)
\(L(\frac12)\) \(\approx\) \(1.740334842\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.87T + 3T^{2} \)
5 \( 1 - 4.26T + 5T^{2} \)
7 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 0.0267T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 + 2.97T + 23T^{2} \)
29 \( 1 + 5.73T + 29T^{2} \)
31 \( 1 - 6.76T + 31T^{2} \)
41 \( 1 + 1.49T + 41T^{2} \)
43 \( 1 - 5.53T + 43T^{2} \)
47 \( 1 - 2.61T + 47T^{2} \)
53 \( 1 - 1.90T + 53T^{2} \)
59 \( 1 - 5.62T + 59T^{2} \)
61 \( 1 - 5.74T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 + 9.27T + 71T^{2} \)
73 \( 1 + 16.2T + 73T^{2} \)
79 \( 1 - 1.41T + 79T^{2} \)
83 \( 1 + 1.69T + 83T^{2} \)
89 \( 1 + 8.27T + 89T^{2} \)
97 \( 1 + 6.52T + 97T^{2} \)
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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

−8.901337535898318943157883123425, −8.263681572330843479470054041792, −7.17031419420960295044417635873, −6.37693308169883740454083705562, −5.72821071158618144354371454285, −5.37750276082461771364595160173, −4.32712548064284588072529358773, −2.71459229533343755092926362443, −1.64536106920158994875112925863, −1.10871814044735780767221254251, 1.10871814044735780767221254251, 1.64536106920158994875112925863, 2.71459229533343755092926362443, 4.32712548064284588072529358773, 5.37750276082461771364595160173, 5.72821071158618144354371454285, 6.37693308169883740454083705562, 7.17031419420960295044417635873, 8.263681572330843479470054041792, 8.901337535898318943157883123425

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