Properties

Label 2-7448-1.1-c1-0-30
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  + 0.684·3-s − 3.65·5-s − 2.53·9-s + 2.37·11-s + 5.46·13-s − 2.49·15-s − 4.86·17-s + 19-s − 1.77·23-s + 8.32·25-s − 3.78·27-s − 6.04·29-s + 1.69·31-s + 1.62·33-s + 5.62·37-s + 3.74·39-s − 2·41-s + 3.10·43-s + 9.24·45-s − 5.23·47-s − 3.32·51-s − 6.46·53-s − 8.67·55-s + 0.684·57-s − 7.52·59-s − 7.75·61-s − 19.9·65-s + ⋯
L(s)  = 1  + 0.395·3-s − 1.63·5-s − 0.843·9-s + 0.716·11-s + 1.51·13-s − 0.644·15-s − 1.17·17-s + 0.229·19-s − 0.370·23-s + 1.66·25-s − 0.728·27-s − 1.12·29-s + 0.305·31-s + 0.283·33-s + 0.924·37-s + 0.599·39-s − 0.312·41-s + 0.474·43-s + 1.37·45-s − 0.762·47-s − 0.465·51-s − 0.888·53-s − 1.17·55-s + 0.0906·57-s − 0.979·59-s − 0.993·61-s − 2.47·65-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

\(L(1)\) \(\approx\) \(1.218251466\)
\(L(\frac12)\) \(\approx\) \(1.218251466\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.684T + 3T^{2} \)
5 \( 1 + 3.65T + 5T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 4.86T + 17T^{2} \)
23 \( 1 + 1.77T + 23T^{2} \)
29 \( 1 + 6.04T + 29T^{2} \)
31 \( 1 - 1.69T + 31T^{2} \)
37 \( 1 - 5.62T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 3.10T + 43T^{2} \)
47 \( 1 + 5.23T + 47T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + 7.52T + 59T^{2} \)
61 \( 1 + 7.75T + 61T^{2} \)
67 \( 1 + 2.62T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 0.961T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + 6.26T + 97T^{2} \)
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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

−7.987037090231207141353050928057, −7.39268154447430809749593856834, −6.46130238287004168272051565809, −6.02089106258964648570404283176, −4.86875749325943434034403631676, −4.08405186118381637894811546634, −3.62989050889987327342366608839, −2.97025566599721680851085903937, −1.77318043839249126537234618105, −0.53934515358848748387159892336, 0.53934515358848748387159892336, 1.77318043839249126537234618105, 2.97025566599721680851085903937, 3.62989050889987327342366608839, 4.08405186118381637894811546634, 4.86875749325943434034403631676, 6.02089106258964648570404283176, 6.46130238287004168272051565809, 7.39268154447430809749593856834, 7.987037090231207141353050928057

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