Properties

Label 2-7448-1.1-c1-0-100
Degree $2$
Conductor $7448$
Sign $-1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 3·9-s − 11-s + 6·17-s + 19-s − 3·23-s + 4·25-s + 6·29-s − 2·31-s − 6·41-s + 9·43-s + 9·45-s + 3·47-s − 2·53-s + 3·55-s + 12·59-s − 11·61-s − 8·67-s + 6·71-s + 15·73-s − 12·79-s + 9·81-s − 7·83-s − 18·85-s + 8·89-s − 3·95-s + 6·97-s + ⋯
L(s)  = 1  − 1.34·5-s − 9-s − 0.301·11-s + 1.45·17-s + 0.229·19-s − 0.625·23-s + 4/5·25-s + 1.11·29-s − 0.359·31-s − 0.937·41-s + 1.37·43-s + 1.34·45-s + 0.437·47-s − 0.274·53-s + 0.404·55-s + 1.56·59-s − 1.40·61-s − 0.977·67-s + 0.712·71-s + 1.75·73-s − 1.35·79-s + 81-s − 0.768·83-s − 1.95·85-s + 0.847·89-s − 0.307·95-s + 0.609·97-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: yes
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 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.73399562032679455937365894379, −7.02028215244095049263424732329, −6.07078470326296157609746681334, −5.45274598721425447175132701183, −4.68576495790058444391416572636, −3.79727198485076033745307334702, −3.26064805881889257533029407709, −2.46158077286027324420675407151, −1.03611344715743476261895002874, 0, 1.03611344715743476261895002874, 2.46158077286027324420675407151, 3.26064805881889257533029407709, 3.79727198485076033745307334702, 4.68576495790058444391416572636, 5.45274598721425447175132701183, 6.07078470326296157609746681334, 7.02028215244095049263424732329, 7.73399562032679455937365894379

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