Properties

Label 2-7623-1.1-c1-0-150
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.25·2-s + 3.10·4-s − 1.54·5-s − 7-s − 2.49·8-s + 3.48·10-s + 2.53·13-s + 2.25·14-s − 0.568·16-s + 6.59·17-s − 7.56·19-s − 4.79·20-s + 5.61·23-s − 2.61·25-s − 5.73·26-s − 3.10·28-s + 5.45·29-s − 5.32·31-s + 6.27·32-s − 14.9·34-s + 1.54·35-s − 6.18·37-s + 17.0·38-s + 3.85·40-s − 9.78·41-s + 9.61·43-s − 12.6·46-s + ⋯
L(s)  = 1  − 1.59·2-s + 1.55·4-s − 0.690·5-s − 0.377·7-s − 0.882·8-s + 1.10·10-s + 0.703·13-s + 0.603·14-s − 0.142·16-s + 1.59·17-s − 1.73·19-s − 1.07·20-s + 1.17·23-s − 0.523·25-s − 1.12·26-s − 0.586·28-s + 1.01·29-s − 0.956·31-s + 1.10·32-s − 2.55·34-s + 0.260·35-s − 1.01·37-s + 2.77·38-s + 0.609·40-s − 1.52·41-s + 1.46·43-s − 1.87·46-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: \(1\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 2.25T + 2T^{2} \)
5 \( 1 + 1.54T + 5T^{2} \)
13 \( 1 - 2.53T + 13T^{2} \)
17 \( 1 - 6.59T + 17T^{2} \)
19 \( 1 + 7.56T + 19T^{2} \)
23 \( 1 - 5.61T + 23T^{2} \)
29 \( 1 - 5.45T + 29T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 + 6.18T + 37T^{2} \)
41 \( 1 + 9.78T + 41T^{2} \)
43 \( 1 - 9.61T + 43T^{2} \)
47 \( 1 - 3.84T + 47T^{2} \)
53 \( 1 - 0.531T + 53T^{2} \)
59 \( 1 + 1.76T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + 9.60T + 67T^{2} \)
71 \( 1 + 9.94T + 71T^{2} \)
73 \( 1 - 9.71T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 + 0.0442T + 83T^{2} \)
89 \( 1 - 6.38T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−7.65082968535845967740609686692, −7.13832168230232769030667681787, −6.43339945054398281088407960742, −5.69731354622826087093961018185, −4.62200080883352055391608988762, −3.73096187131031355903018009933, −2.99389245611146487024444685193, −1.89544040996673538445174186295, −0.998397357300487989738796887830, 0, 0.998397357300487989738796887830, 1.89544040996673538445174186295, 2.99389245611146487024444685193, 3.73096187131031355903018009933, 4.62200080883352055391608988762, 5.69731354622826087093961018185, 6.43339945054398281088407960742, 7.13832168230232769030667681787, 7.65082968535845967740609686692

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