Properties

Label 2-7448-1.1-c1-0-182
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.41·3-s − 5-s + 8.65·9-s − 3.24·11-s − 3.41·13-s − 3.41·15-s − 6.82·17-s − 19-s + 4.41·23-s − 4·25-s + 19.3·27-s − 9.41·29-s + 4.58·31-s − 11.0·33-s − 6.58·37-s − 11.6·39-s − 5.65·41-s − 3.24·43-s − 8.65·45-s − 1.24·47-s − 23.3·51-s − 5.41·53-s + 3.24·55-s − 3.41·57-s − 0.828·59-s − 5·61-s + 3.41·65-s + ⋯
L(s)  = 1  + 1.97·3-s − 0.447·5-s + 2.88·9-s − 0.977·11-s − 0.946·13-s − 0.881·15-s − 1.65·17-s − 0.229·19-s + 0.920·23-s − 0.800·25-s + 3.71·27-s − 1.74·29-s + 0.823·31-s − 1.92·33-s − 1.08·37-s − 1.86·39-s − 0.883·41-s − 0.494·43-s − 1.29·45-s − 0.181·47-s − 3.26·51-s − 0.743·53-s + 0.437·55-s − 0.452·57-s − 0.107·59-s − 0.640·61-s + 0.423·65-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: \(1\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 6.82T + 17T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + 9.41T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + 5.65T + 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + 1.24T + 47T^{2} \)
53 \( 1 + 5.41T + 53T^{2} \)
59 \( 1 + 0.828T + 59T^{2} \)
61 \( 1 + 5T + 61T^{2} \)
67 \( 1 - 3.17T + 67T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 3.89T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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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

−7.68671780457334911703013126390, −7.17083149056855568931119015950, −6.50174383654703528385856391609, −5.07988392016100512362776925541, −4.56785945014152545214228065817, −3.75173963786080742346754762413, −3.07867313043376728731086171348, −2.32071188980507502358767702133, −1.77882389418539504067161352131, 0, 1.77882389418539504067161352131, 2.32071188980507502358767702133, 3.07867313043376728731086171348, 3.75173963786080742346754762413, 4.56785945014152545214228065817, 5.07988392016100512362776925541, 6.50174383654703528385856391609, 7.17083149056855568931119015950, 7.68671780457334911703013126390

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