Properties

Label 2-7623-1.1-c1-0-90
Degree $2$
Conductor $7623$
Sign $1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s − 4·13-s + 14-s − 16-s + 4·17-s − 2·20-s + 4·23-s − 25-s − 4·26-s − 28-s − 6·29-s + 10·31-s + 5·32-s + 4·34-s + 2·35-s − 6·37-s − 6·40-s + 4·41-s − 12·43-s + 4·46-s + 10·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s − 0.447·20-s + 0.834·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s + 1.79·31-s + 0.883·32-s + 0.685·34-s + 0.338·35-s − 0.986·37-s − 0.948·40-s + 0.624·41-s − 1.82·43-s + 0.589·46-s + 1.45·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(60.8699\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.769521394\)
\(L(\frac12)\) \(\approx\) \(2.769521394\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.87739468442856243522484593458, −7.07007000345216400828661927108, −6.31251542982324707519501957045, −5.45217811977635791883277118514, −5.22894511431647767966156393918, −4.45772251821753336888632495633, −3.58922471493276128945836920884, −2.78349816157540378309057592549, −1.96167877918167104055842099400, −0.75006805308184274174858297412, 0.75006805308184274174858297412, 1.96167877918167104055842099400, 2.78349816157540378309057592549, 3.58922471493276128945836920884, 4.45772251821753336888632495633, 5.22894511431647767966156393918, 5.45217811977635791883277118514, 6.31251542982324707519501957045, 7.07007000345216400828661927108, 7.87739468442856243522484593458

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