Properties

Label 2-7448-1.1-c1-0-138
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
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  + 2.04·3-s + 3.17·5-s + 1.16·9-s + 2.64·11-s + 5.60·13-s + 6.48·15-s + 7.48·17-s − 19-s + 4.87·23-s + 5.09·25-s − 3.74·27-s + 2.60·29-s − 3.41·31-s + 5.39·33-s − 9.05·37-s + 11.4·39-s − 10.4·41-s + 3.77·43-s + 3.70·45-s + 7.21·47-s + 15.2·51-s − 9.74·53-s + 8.39·55-s − 2.04·57-s + 3.30·59-s − 2.35·61-s + 17.7·65-s + ⋯
L(s)  = 1  + 1.17·3-s + 1.42·5-s + 0.389·9-s + 0.796·11-s + 1.55·13-s + 1.67·15-s + 1.81·17-s − 0.229·19-s + 1.01·23-s + 1.01·25-s − 0.719·27-s + 0.484·29-s − 0.612·31-s + 0.938·33-s − 1.48·37-s + 1.83·39-s − 1.62·41-s + 0.575·43-s + 0.552·45-s + 1.05·47-s + 2.13·51-s − 1.33·53-s + 1.13·55-s − 0.270·57-s + 0.430·59-s − 0.301·61-s + 2.20·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7448\)    =    \(2^{3} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(59.4725\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.195184070\)
\(L(\frac12)\) \(\approx\) \(5.195184070\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 2.04T + 3T^{2} \)
5 \( 1 - 3.17T + 5T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 - 7.48T + 17T^{2} \)
23 \( 1 - 4.87T + 23T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 + 9.05T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 3.77T + 43T^{2} \)
47 \( 1 - 7.21T + 47T^{2} \)
53 \( 1 + 9.74T + 53T^{2} \)
59 \( 1 - 3.30T + 59T^{2} \)
61 \( 1 + 2.35T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 - 8.57T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 + 6.51T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + 7.06T + 97T^{2} \)
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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

−8.184123498515118066912100537527, −7.14478996066963123021458571394, −6.55449264172348129969144627031, −5.68758148742881210304473925446, −5.37131210403819024240035216620, −4.04353688401891213443012804667, −3.36999502093772818612884992383, −2.81327218714800778804683842661, −1.66597776783543413463773721574, −1.29523553213946620977412569662, 1.29523553213946620977412569662, 1.66597776783543413463773721574, 2.81327218714800778804683842661, 3.36999502093772818612884992383, 4.04353688401891213443012804667, 5.37131210403819024240035216620, 5.68758148742881210304473925446, 6.55449264172348129969144627031, 7.14478996066963123021458571394, 8.184123498515118066912100537527

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