Properties

Label 2-7623-1.1-c1-0-79
Degree $2$
Conductor $7623$
Sign $1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
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  − 0.871·2-s − 1.24·4-s + 4.06·5-s − 7-s + 2.82·8-s − 3.54·10-s − 6.09·13-s + 0.871·14-s + 0.0189·16-s − 1.70·17-s + 5.45·19-s − 5.04·20-s + 6.39·23-s + 11.5·25-s + 5.30·26-s + 1.24·28-s − 4.74·29-s − 4.36·31-s − 5.66·32-s + 1.48·34-s − 4.06·35-s + 2.43·37-s − 4.75·38-s + 11.4·40-s + 3.21·41-s + 0.127·43-s − 5.57·46-s + ⋯
L(s)  = 1  − 0.616·2-s − 0.620·4-s + 1.81·5-s − 0.377·7-s + 0.998·8-s − 1.12·10-s − 1.68·13-s + 0.232·14-s + 0.00474·16-s − 0.412·17-s + 1.25·19-s − 1.12·20-s + 1.33·23-s + 2.30·25-s + 1.04·26-s + 0.234·28-s − 0.881·29-s − 0.783·31-s − 1.00·32-s + 0.254·34-s − 0.687·35-s + 0.401·37-s − 0.771·38-s + 1.81·40-s + 0.501·41-s + 0.0193·43-s − 0.821·46-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.530515535\)
\(L(\frac12)\) \(\approx\) \(1.530515535\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.871T + 2T^{2} \)
5 \( 1 - 4.06T + 5T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 + 1.70T + 17T^{2} \)
19 \( 1 - 5.45T + 19T^{2} \)
23 \( 1 - 6.39T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 + 4.36T + 31T^{2} \)
37 \( 1 - 2.43T + 37T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 - 0.127T + 43T^{2} \)
47 \( 1 + 8.46T + 47T^{2} \)
53 \( 1 - 4.71T + 53T^{2} \)
59 \( 1 + 4.63T + 59T^{2} \)
61 \( 1 - 5.31T + 61T^{2} \)
67 \( 1 - 14.0T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 - 6.27T + 79T^{2} \)
83 \( 1 - 0.127T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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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

−7.84718036814870635729665745005, −7.19589381051116622804622758423, −6.63729691271786958667295319307, −5.56065153958213317893384722786, −5.23593675514092614152812105745, −4.59179736809752794807891268087, −3.34100362963950759654337721775, −2.47474818131028224046092608900, −1.72325418614616077212802393912, −0.69479723325253449137532723031, 0.69479723325253449137532723031, 1.72325418614616077212802393912, 2.47474818131028224046092608900, 3.34100362963950759654337721775, 4.59179736809752794807891268087, 5.23593675514092614152812105745, 5.56065153958213317893384722786, 6.63729691271786958667295319307, 7.19589381051116622804622758423, 7.84718036814870635729665745005

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