Properties

Label 2-7448-1.1-c1-0-174
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  + 2·3-s + 5-s + 9-s + 2.27·11-s − 2·13-s + 2·15-s − 7.27·17-s + 19-s − 7.54·23-s − 4·25-s − 4·27-s − 4·29-s + 4·31-s + 4.54·33-s + 2·37-s − 4·39-s + 4.54·41-s − 43-s + 45-s + 2.27·47-s − 14.5·51-s − 6.54·53-s + 2.27·55-s + 2·57-s + 4·59-s − 12.2·61-s − 2·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.333·9-s + 0.685·11-s − 0.554·13-s + 0.516·15-s − 1.76·17-s + 0.229·19-s − 1.57·23-s − 0.800·25-s − 0.769·27-s − 0.742·29-s + 0.718·31-s + 0.792·33-s + 0.328·37-s − 0.640·39-s + 0.710·41-s − 0.152·43-s + 0.149·45-s + 0.331·47-s − 2.03·51-s − 0.899·53-s + 0.306·55-s + 0.264·57-s + 0.520·59-s − 1.57·61-s − 0.248·65-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 \( 1 - 2T + 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
23 \( 1 + 7.54T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 - 2.27T + 47T^{2} \)
53 \( 1 + 6.54T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 3.72T + 73T^{2} \)
79 \( 1 - 0.549T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 8.54T + 97T^{2} \)
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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

−7.71714917823861638993109811044, −6.90859687520260318448077582540, −6.20283407147173623240099769954, −5.55528799533582143705735827223, −4.35780900733287766682040458910, −4.03452564750507730897093142648, −2.97466419594173721666056639922, −2.25742215999456096568911350087, −1.68847621078451017832376543411, 0, 1.68847621078451017832376543411, 2.25742215999456096568911350087, 2.97466419594173721666056639922, 4.03452564750507730897093142648, 4.35780900733287766682040458910, 5.55528799533582143705735827223, 6.20283407147173623240099769954, 6.90859687520260318448077582540, 7.71714917823861638993109811044

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