Properties

Label 2-7448-1.1-c1-0-161
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  + 2.62·3-s − 2.40·5-s + 3.90·9-s + 0.814·11-s − 6.10·13-s − 6.32·15-s − 1.76·17-s − 19-s + 7.28·23-s + 0.794·25-s + 2.38·27-s + 8.27·29-s + 6.60·31-s + 2.14·33-s + 3.48·37-s − 16.0·39-s − 12.4·41-s − 6.29·43-s − 9.40·45-s − 7.68·47-s − 4.62·51-s + 0.143·53-s − 1.96·55-s − 2.62·57-s − 9.50·59-s + 3.68·61-s + 14.6·65-s + ⋯
L(s)  = 1  + 1.51·3-s − 1.07·5-s + 1.30·9-s + 0.245·11-s − 1.69·13-s − 1.63·15-s − 0.426·17-s − 0.229·19-s + 1.51·23-s + 0.158·25-s + 0.458·27-s + 1.53·29-s + 1.18·31-s + 0.372·33-s + 0.572·37-s − 2.56·39-s − 1.94·41-s − 0.959·43-s − 1.40·45-s − 1.12·47-s − 0.647·51-s + 0.0197·53-s − 0.264·55-s − 0.348·57-s − 1.23·59-s + 0.471·61-s + 1.82·65-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: Trivial
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 \( 1 - 2.62T + 3T^{2} \)
5 \( 1 + 2.40T + 5T^{2} \)
11 \( 1 - 0.814T + 11T^{2} \)
13 \( 1 + 6.10T + 13T^{2} \)
17 \( 1 + 1.76T + 17T^{2} \)
23 \( 1 - 7.28T + 23T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 - 6.60T + 31T^{2} \)
37 \( 1 - 3.48T + 37T^{2} \)
41 \( 1 + 12.4T + 41T^{2} \)
43 \( 1 + 6.29T + 43T^{2} \)
47 \( 1 + 7.68T + 47T^{2} \)
53 \( 1 - 0.143T + 53T^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 + 2.28T + 67T^{2} \)
71 \( 1 - 2.05T + 71T^{2} \)
73 \( 1 - 2.01T + 73T^{2} \)
79 \( 1 + 5.85T + 79T^{2} \)
83 \( 1 + 1.76T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−7.75110160705905858724641412050, −6.97099152798535585554421751014, −6.60186875864970344697038743299, −5.02348460819135264140331093080, −4.62473632563215661563153660863, −3.81955395799156122817363021318, −2.95605182937106734468023450883, −2.63363311433670156875393507634, −1.45052122443862501107392573360, 0, 1.45052122443862501107392573360, 2.63363311433670156875393507634, 2.95605182937106734468023450883, 3.81955395799156122817363021318, 4.62473632563215661563153660863, 5.02348460819135264140331093080, 6.60186875864970344697038743299, 6.97099152798535585554421751014, 7.75110160705905858724641412050

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