Properties

Label 2-7448-1.1-c1-0-109
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  + 2.39·3-s + 3.25·5-s + 2.71·9-s − 0.402·11-s − 1.63·13-s + 7.77·15-s + 4.43·17-s + 19-s + 0.837·23-s + 5.57·25-s − 0.686·27-s − 5.33·29-s + 3.16·31-s − 0.962·33-s + 11.4·37-s − 3.89·39-s − 2·41-s + 1.93·43-s + 8.82·45-s + 12.6·47-s + 10.6·51-s − 11.5·53-s − 1.31·55-s + 2.39·57-s + 5.16·59-s + 0.126·61-s − 5.30·65-s + ⋯
L(s)  = 1  + 1.37·3-s + 1.45·5-s + 0.904·9-s − 0.121·11-s − 0.452·13-s + 2.00·15-s + 1.07·17-s + 0.229·19-s + 0.174·23-s + 1.11·25-s − 0.132·27-s − 0.990·29-s + 0.567·31-s − 0.167·33-s + 1.87·37-s − 0.623·39-s − 0.312·41-s + 0.295·43-s + 1.31·45-s + 1.84·47-s + 1.48·51-s − 1.58·53-s − 0.176·55-s + 0.316·57-s + 0.673·59-s + 0.0161·61-s − 0.657·65-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

\(L(1)\) \(\approx\) \(4.869301916\)
\(L(\frac12)\) \(\approx\) \(4.869301916\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.39T + 3T^{2} \)
5 \( 1 - 3.25T + 5T^{2} \)
11 \( 1 + 0.402T + 11T^{2} \)
13 \( 1 + 1.63T + 13T^{2} \)
17 \( 1 - 4.43T + 17T^{2} \)
23 \( 1 - 0.837T + 23T^{2} \)
29 \( 1 + 5.33T + 29T^{2} \)
31 \( 1 - 3.16T + 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 1.93T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 5.16T + 59T^{2} \)
61 \( 1 - 0.126T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + 2.17T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 + 4.40T + 79T^{2} \)
83 \( 1 + 1.87T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 + 7.50T + 97T^{2} \)
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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

−7.78519710194102977745620684544, −7.52388425241734352968599974593, −6.46593618180412492942795526900, −5.79059626227410906473786155328, −5.18176454034324390915599974195, −4.19770514933298373942048361717, −3.29207584995394540115068442489, −2.60359025475657129036670078771, −2.05781482670116854294927745620, −1.09802029650983048767998412188, 1.09802029650983048767998412188, 2.05781482670116854294927745620, 2.60359025475657129036670078771, 3.29207584995394540115068442489, 4.19770514933298373942048361717, 5.18176454034324390915599974195, 5.79059626227410906473786155328, 6.46593618180412492942795526900, 7.52388425241734352968599974593, 7.78519710194102977745620684544

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