Properties

Label 2-7623-1.1-c1-0-91
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  − 1.60·2-s + 0.568·4-s + 1.76·5-s + 7-s + 2.29·8-s − 2.82·10-s − 0.237·13-s − 1.60·14-s − 4.81·16-s + 6.14·17-s − 1.53·19-s + 1.00·20-s + 3.89·23-s − 1.88·25-s + 0.381·26-s + 0.568·28-s − 4.45·29-s + 8.52·31-s + 3.12·32-s − 9.84·34-s + 1.76·35-s − 3.64·37-s + 2.46·38-s + 4.04·40-s − 1.90·41-s + 11.7·43-s − 6.24·46-s + ⋯
L(s)  = 1  − 1.13·2-s + 0.284·4-s + 0.789·5-s + 0.377·7-s + 0.811·8-s − 0.894·10-s − 0.0659·13-s − 0.428·14-s − 1.20·16-s + 1.48·17-s − 0.352·19-s + 0.224·20-s + 0.812·23-s − 0.377·25-s + 0.0747·26-s + 0.107·28-s − 0.827·29-s + 1.53·31-s + 0.552·32-s − 1.68·34-s + 0.298·35-s − 0.599·37-s + 0.399·38-s + 0.640·40-s − 0.297·41-s + 1.78·43-s − 0.920·46-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

\(L(1)\) \(\approx\) \(1.367717440\)
\(L(\frac12)\) \(\approx\) \(1.367717440\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.60T + 2T^{2} \)
5 \( 1 - 1.76T + 5T^{2} \)
13 \( 1 + 0.237T + 13T^{2} \)
17 \( 1 - 6.14T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + 4.45T + 29T^{2} \)
31 \( 1 - 8.52T + 31T^{2} \)
37 \( 1 + 3.64T + 37T^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 - 3.14T + 53T^{2} \)
59 \( 1 + 2.56T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + 7.37T + 71T^{2} \)
73 \( 1 + 4.60T + 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 + 4.68T + 97T^{2} \)
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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

−7.958204183437005377414205077542, −7.44842589081881940192621205298, −6.64661489343800379232518273162, −5.81649486042201298984575287817, −5.16271082513793405273910871704, −4.41397805421914120664203374383, −3.40912450308021631319884280348, −2.35378360969816408767060792244, −1.54883400628448331681135671755, −0.75046854305491917465400561184, 0.75046854305491917465400561184, 1.54883400628448331681135671755, 2.35378360969816408767060792244, 3.40912450308021631319884280348, 4.41397805421914120664203374383, 5.16271082513793405273910871704, 5.81649486042201298984575287817, 6.64661489343800379232518273162, 7.44842589081881940192621205298, 7.958204183437005377414205077542

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