Properties

Label 2-7448-1.1-c1-0-36
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  + 0.912·3-s − 1.66·5-s − 2.16·9-s − 0.242·11-s − 5.76·13-s − 1.51·15-s + 5.76·17-s + 19-s + 6.61·23-s − 2.22·25-s − 4.71·27-s + 8.91·29-s − 5.93·31-s − 0.220·33-s − 9.80·37-s − 5.26·39-s − 2·41-s − 7.83·43-s + 3.60·45-s + 1.57·47-s + 5.26·51-s + 6.48·53-s + 0.403·55-s + 0.912·57-s + 1.07·59-s + 9.10·61-s + 9.60·65-s + ⋯
L(s)  = 1  + 0.526·3-s − 0.744·5-s − 0.722·9-s − 0.0730·11-s − 1.60·13-s − 0.392·15-s + 1.39·17-s + 0.229·19-s + 1.37·23-s − 0.445·25-s − 0.907·27-s + 1.65·29-s − 1.06·31-s − 0.0384·33-s − 1.61·37-s − 0.842·39-s − 0.312·41-s − 1.19·43-s + 0.538·45-s + 0.230·47-s + 0.736·51-s + 0.890·53-s + 0.0543·55-s + 0.120·57-s + 0.139·59-s + 1.16·61-s + 1.19·65-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

\(L(1)\) \(\approx\) \(1.444294550\)
\(L(\frac12)\) \(\approx\) \(1.444294550\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.912T + 3T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
11 \( 1 + 0.242T + 11T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 - 5.76T + 17T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + 5.93T + 31T^{2} \)
37 \( 1 + 9.80T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7.83T + 43T^{2} \)
47 \( 1 - 1.57T + 47T^{2} \)
53 \( 1 - 6.48T + 53T^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 - 9.10T + 61T^{2} \)
67 \( 1 - 8.02T + 67T^{2} \)
71 \( 1 - 1.22T + 71T^{2} \)
73 \( 1 - 0.801T + 73T^{2} \)
79 \( 1 + 7.58T + 79T^{2} \)
83 \( 1 + 8.91T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 1.11T + 97T^{2} \)
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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

−7.980742566578950828120203936553, −7.19894385785835669691935238223, −6.85853974334204954132604138828, −5.44592272442077110596749590912, −5.27158092336027634226410497901, −4.25827484856800844191050663282, −3.29250820079081998916134045192, −2.95743450417319163042219427380, −1.91874588274641452236736494683, −0.56757633580984542772623305889, 0.56757633580984542772623305889, 1.91874588274641452236736494683, 2.95743450417319163042219427380, 3.29250820079081998916134045192, 4.25827484856800844191050663282, 5.27158092336027634226410497901, 5.44592272442077110596749590912, 6.85853974334204954132604138828, 7.19894385785835669691935238223, 7.980742566578950828120203936553

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