Properties

Label 2-7448-1.1-c1-0-114
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  + 2.85·3-s + 0.827·5-s + 5.16·9-s + 5.71·11-s − 0.0340·13-s + 2.36·15-s − 4.23·17-s + 19-s + 4.89·23-s − 4.31·25-s + 6.17·27-s + 9.29·29-s − 2.53·31-s + 16.3·33-s + 4.58·37-s − 0.0972·39-s − 2·41-s + 9.61·43-s + 4.27·45-s − 11.3·47-s − 12.0·51-s + 3.43·53-s + 4.73·55-s + 2.85·57-s − 7.35·59-s + 2.49·61-s − 0.0281·65-s + ⋯
L(s)  = 1  + 1.64·3-s + 0.370·5-s + 1.72·9-s + 1.72·11-s − 0.00943·13-s + 0.610·15-s − 1.02·17-s + 0.229·19-s + 1.02·23-s − 0.862·25-s + 1.18·27-s + 1.72·29-s − 0.455·31-s + 2.84·33-s + 0.753·37-s − 0.0155·39-s − 0.312·41-s + 1.46·43-s + 0.636·45-s − 1.65·47-s − 1.69·51-s + 0.471·53-s + 0.638·55-s + 0.378·57-s − 0.957·59-s + 0.319·61-s − 0.00349·65-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

\(L(1)\) \(\approx\) \(4.923994750\)
\(L(\frac12)\) \(\approx\) \(4.923994750\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.85T + 3T^{2} \)
5 \( 1 - 0.827T + 5T^{2} \)
11 \( 1 - 5.71T + 11T^{2} \)
13 \( 1 + 0.0340T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 9.29T + 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 - 4.58T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 9.61T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 3.43T + 53T^{2} \)
59 \( 1 + 7.35T + 59T^{2} \)
61 \( 1 - 2.49T + 61T^{2} \)
67 \( 1 + 6.98T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 0.751T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 - 8.49T + 97T^{2} \)
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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

−8.008110816422012422727281036918, −7.26193332135365972439442855003, −6.64077368345259466565547078637, −6.03577564921452833172576841461, −4.75559332373637974660815777526, −4.18321580807067416901304850295, −3.45106202508690602210847041815, −2.72069388028561751606906667875, −1.92100403643489553231850075471, −1.12376992463193954143160693599, 1.12376992463193954143160693599, 1.92100403643489553231850075471, 2.72069388028561751606906667875, 3.45106202508690602210847041815, 4.18321580807067416901304850295, 4.75559332373637974660815777526, 6.03577564921452833172576841461, 6.64077368345259466565547078637, 7.26193332135365972439442855003, 8.008110816422012422727281036918

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