Properties

Label 2-7616-1.1-c1-0-55
Degree $2$
Conductor $7616$
Sign $1$
Analytic cond. $60.8140$
Root an. cond. $7.79833$
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  + 0.618·3-s + 1.61·5-s + 7-s − 2.61·9-s − 5.23·11-s + 3.23·13-s + 1.00·15-s + 17-s + 0.472·19-s + 0.618·21-s − 5.70·23-s − 2.38·25-s − 3.47·27-s − 7.70·29-s + 9.32·31-s − 3.23·33-s + 1.61·35-s + 8.47·37-s + 2.00·39-s + 11.0·41-s − 0.909·43-s − 4.23·45-s − 0.472·47-s + 49-s + 0.618·51-s + 13.7·53-s − 8.47·55-s + ⋯
L(s)  = 1  + 0.356·3-s + 0.723·5-s + 0.377·7-s − 0.872·9-s − 1.57·11-s + 0.897·13-s + 0.258·15-s + 0.242·17-s + 0.108·19-s + 0.134·21-s − 1.19·23-s − 0.476·25-s − 0.668·27-s − 1.43·29-s + 1.67·31-s − 0.563·33-s + 0.273·35-s + 1.39·37-s + 0.320·39-s + 1.73·41-s − 0.138·43-s − 0.631·45-s − 0.0688·47-s + 0.142·49-s + 0.0865·51-s + 1.89·53-s − 1.14·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7616\)    =    \(2^{6} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(60.8140\)
Root analytic conductor: \(7.79833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7616,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.271504393\)
\(L(\frac12)\) \(\approx\) \(2.271504393\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good3 \( 1 - 0.618T + 3T^{2} \)
5 \( 1 - 1.61T + 5T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
19 \( 1 - 0.472T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 + 7.70T + 29T^{2} \)
31 \( 1 - 9.32T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 0.909T + 43T^{2} \)
47 \( 1 + 0.472T + 47T^{2} \)
53 \( 1 - 13.7T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.32T + 61T^{2} \)
67 \( 1 + 1.90T + 67T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 3.90T + 97T^{2} \)
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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

−7.974224657931626322975793193405, −7.45952258110471161977219170812, −6.10743365925654954095622486285, −5.91475206560562586175191756142, −5.23995929346021743976510940407, −4.30136791525254213536169601384, −3.43135527213051253590442640954, −2.49338630976254685347299399477, −2.09183037419661615297929887122, −0.71719957243397953576759096319, 0.71719957243397953576759096319, 2.09183037419661615297929887122, 2.49338630976254685347299399477, 3.43135527213051253590442640954, 4.30136791525254213536169601384, 5.23995929346021743976510940407, 5.91475206560562586175191756142, 6.10743365925654954095622486285, 7.45952258110471161977219170812, 7.974224657931626322975793193405

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