Properties

Label 2-7448-1.1-c1-0-26
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  − 2.46·3-s − 4.03·5-s + 3.10·9-s + 1.60·11-s + 3.15·13-s + 9.97·15-s + 0.652·17-s + 19-s + 2.91·23-s + 11.3·25-s − 0.248·27-s + 2.12·29-s − 7.02·31-s − 3.95·33-s − 5.30·37-s − 7.79·39-s − 2·41-s + 7.97·43-s − 12.5·45-s − 0.605·47-s − 1.61·51-s + 12.4·53-s − 6.47·55-s − 2.46·57-s + 14.4·59-s + 9.34·61-s − 12.7·65-s + ⋯
L(s)  = 1  − 1.42·3-s − 1.80·5-s + 1.03·9-s + 0.483·11-s + 0.874·13-s + 2.57·15-s + 0.158·17-s + 0.229·19-s + 0.608·23-s + 2.26·25-s − 0.0477·27-s + 0.393·29-s − 1.26·31-s − 0.688·33-s − 0.872·37-s − 1.24·39-s − 0.312·41-s + 1.21·43-s − 1.86·45-s − 0.0883·47-s − 0.225·51-s + 1.71·53-s − 0.872·55-s − 0.327·57-s + 1.88·59-s + 1.19·61-s − 1.58·65-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

\(L(1)\) \(\approx\) \(0.6718498079\)
\(L(\frac12)\) \(\approx\) \(0.6718498079\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 3T^{2} \)
5 \( 1 + 4.03T + 5T^{2} \)
11 \( 1 - 1.60T + 11T^{2} \)
13 \( 1 - 3.15T + 13T^{2} \)
17 \( 1 - 0.652T + 17T^{2} \)
23 \( 1 - 2.91T + 23T^{2} \)
29 \( 1 - 2.12T + 29T^{2} \)
31 \( 1 + 7.02T + 31T^{2} \)
37 \( 1 + 5.30T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 7.97T + 43T^{2} \)
47 \( 1 + 0.605T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 9.34T + 61T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 4.85T + 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 + 2.27T + 89T^{2} \)
97 \( 1 - 1.24T + 97T^{2} \)
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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

−7.71956834490964933757300386331, −7.04072708913272417851923800090, −6.68836722709374113914995997174, −5.64930909257193306321698241465, −5.21136181181800599848132520073, −4.16688407413964735900730539683, −3.90862994319465995814340749311, −2.93966823292409946157197193714, −1.29095645664281311727707889651, −0.50755109545045332146904247087, 0.50755109545045332146904247087, 1.29095645664281311727707889651, 2.93966823292409946157197193714, 3.90862994319465995814340749311, 4.16688407413964735900730539683, 5.21136181181800599848132520073, 5.64930909257193306321698241465, 6.68836722709374113914995997174, 7.04072708913272417851923800090, 7.71956834490964933757300386331

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