Properties

Label 2-7098-1.1-c1-0-50
Degree $2$
Conductor $7098$
Sign $1$
Analytic cond. $56.6778$
Root an. cond. $7.52846$
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-s + 3-s + 4-s − 3.46·5-s − 6-s + 7-s − 8-s + 9-s + 3.46·10-s + 4·11-s + 12-s − 14-s − 3.46·15-s + 16-s + 6.46·17-s − 18-s + 7.46·19-s − 3.46·20-s + 21-s − 4·22-s + 5.73·23-s − 24-s + 6.99·25-s + 27-s + 28-s + 8·29-s + 3.46·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.54·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 1.20·11-s + 0.288·12-s − 0.267·14-s − 0.894·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s + 1.71·19-s − 0.774·20-s + 0.218·21-s − 0.852·22-s + 1.19·23-s − 0.204·24-s + 1.39·25-s + 0.192·27-s + 0.188·28-s + 1.48·29-s + 0.632·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7098\)    =    \(2 \cdot 3 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(56.6778\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7098} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.850241259\)
\(L(\frac12)\) \(\approx\) \(1.850241259\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 - 7.46T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 4.92T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 4.46T + 43T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 + 8.26T + 53T^{2} \)
59 \( 1 - 6.26T + 59T^{2} \)
61 \( 1 + 5.19T + 61T^{2} \)
67 \( 1 + 7.19T + 67T^{2} \)
71 \( 1 - 1.92T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 + 9.46T + 79T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−7.908296408879606614790903769136, −7.32138610246219232948122183523, −7.10678371512682751386119663084, −5.93175064712730447794545039555, −4.99576293044204927356687923117, −4.16771682654335727415482345369, −3.34378688986018763777195252395, −3.00869747050779622771990597275, −1.42065742635199413001210269792, −0.859833862515387912114742919619, 0.859833862515387912114742919619, 1.42065742635199413001210269792, 3.00869747050779622771990597275, 3.34378688986018763777195252395, 4.16771682654335727415482345369, 4.99576293044204927356687923117, 5.93175064712730447794545039555, 7.10678371512682751386119663084, 7.32138610246219232948122183523, 7.908296408879606614790903769136

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