Properties

Label 2-7623-1.1-c1-0-32
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  − 2.12·2-s + 2.51·4-s − 0.484·5-s − 7-s − 1.09·8-s + 1.03·10-s − 5.60·13-s + 2.12·14-s − 2.70·16-s + 5.60·17-s + 5.28·19-s − 1.21·20-s − 2.48·23-s − 4.76·25-s + 11.9·26-s − 2.51·28-s + 5.28·29-s − 7.12·31-s + 7.93·32-s − 11.9·34-s + 0.484·35-s − 0.235·37-s − 11.2·38-s + 0.530·40-s + 2.39·41-s − 1.03·43-s + 5.28·46-s + ⋯
L(s)  = 1  − 1.50·2-s + 1.25·4-s − 0.216·5-s − 0.377·7-s − 0.387·8-s + 0.325·10-s − 1.55·13-s + 0.567·14-s − 0.676·16-s + 1.36·17-s + 1.21·19-s − 0.272·20-s − 0.518·23-s − 0.952·25-s + 2.33·26-s − 0.475·28-s + 0.980·29-s − 1.27·31-s + 1.40·32-s − 2.04·34-s + 0.0819·35-s − 0.0386·37-s − 1.82·38-s + 0.0839·40-s + 0.373·41-s − 0.157·43-s + 0.778·46-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

\(L(1)\) \(\approx\) \(0.5532540827\)
\(L(\frac12)\) \(\approx\) \(0.5532540827\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12T + 2T^{2} \)
5 \( 1 + 0.484T + 5T^{2} \)
13 \( 1 + 5.60T + 13T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 - 5.28T + 19T^{2} \)
23 \( 1 + 2.48T + 23T^{2} \)
29 \( 1 - 5.28T + 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 + 0.235T + 37T^{2} \)
41 \( 1 - 2.39T + 41T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 - 3.03T + 53T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 + 2.39T + 73T^{2} \)
79 \( 1 + 9.03T + 79T^{2} \)
83 \( 1 + 3.21T + 83T^{2} \)
89 \( 1 - 1.26T + 89T^{2} \)
97 \( 1 + 8.79T + 97T^{2} \)
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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

−7.80051947495501356004631111890, −7.44044066038661582990948433565, −6.94436913731595214760748151708, −5.87336571580950053796768442086, −5.22634774908598370828258254780, −4.27472614353970426702859079659, −3.27726852256310698547683487435, −2.47145428461977226798772875841, −1.50746858203711461827986552420, −0.48202598319545410468036478655, 0.48202598319545410468036478655, 1.50746858203711461827986552420, 2.47145428461977226798772875841, 3.27726852256310698547683487435, 4.27472614353970426702859079659, 5.22634774908598370828258254780, 5.87336571580950053796768442086, 6.94436913731595214760748151708, 7.44044066038661582990948433565, 7.80051947495501356004631111890

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