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Properties

Label 2-7616-1.1-c1-0-184
Degree $2$
Conductor $7616$
Sign $-1$
Analytic cond. $60.8140$
Root an. cond. $7.79833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 9-s + 6·11-s − 6·13-s − 17-s − 4·19-s + 2·21-s − 4·23-s − 5·25-s − 4·27-s − 4·29-s + 12·33-s − 8·37-s − 12·39-s − 10·41-s + 12·43-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 8·57-s − 4·61-s + 63-s − 4·67-s − 8·69-s − 12·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 1.66·13-s − 0.242·17-s − 0.917·19-s + 0.436·21-s − 0.834·23-s − 25-s − 0.769·27-s − 0.742·29-s + 2.08·33-s − 1.31·37-s − 1.92·39-s − 1.56·41-s + 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 1.05·57-s − 0.512·61-s + 0.125·63-s − 0.488·67-s − 0.963·69-s − 1.42·71-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7616,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.59202693064829458896909111624, −7.02816168447269739369120985265, −6.23181047138306935169264311709, −5.42793887913299596958179370300, −4.34486329867371249957719224218, −4.00174760970107223213727742563, −3.10510435236463766649558282055, −2.12656194077689104630007103595, −1.71186234567213227440845229269, 0, 1.71186234567213227440845229269, 2.12656194077689104630007103595, 3.10510435236463766649558282055, 4.00174760970107223213727742563, 4.34486329867371249957719224218, 5.42793887913299596958179370300, 6.23181047138306935169264311709, 7.02816168447269739369120985265, 7.59202693064829458896909111624

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