Properties

Label 2-7098-1.1-c1-0-40
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 + 2·5-s + 6-s + 7-s − 8-s + 9-s − 2·10-s + 5.17·11-s − 12-s − 14-s − 2·15-s + 16-s + 5·17-s − 18-s − 3.17·19-s + 2·20-s − 21-s − 5.17·22-s − 8.17·23-s + 24-s − 25-s − 27-s + 28-s − 3.17·29-s + 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.632·10-s + 1.56·11-s − 0.288·12-s − 0.267·14-s − 0.516·15-s + 0.250·16-s + 1.21·17-s − 0.235·18-s − 0.729·19-s + 0.447·20-s − 0.218·21-s − 1.10·22-s − 1.70·23-s + 0.204·24-s − 0.200·25-s − 0.192·27-s + 0.188·28-s − 0.590·29-s + 0.365·30-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

\(L(1)\) \(\approx\) \(1.664796518\)
\(L(\frac12)\) \(\approx\) \(1.664796518\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 5T^{2} \)
11 \( 1 - 5.17T + 11T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + 3.17T + 19T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 7.17T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 - 7.17T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 3.82T + 67T^{2} \)
71 \( 1 - 2.17T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 6.17T + 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.83676740257569031736813294144, −7.33133297209796264462224475836, −6.40921079103539980944260610478, −5.92702329900097779158484049030, −5.48475911813519175659179726696, −4.22643559376834830035243990149, −3.71720204573071596317679520498, −2.27941211470107522943892843184, −1.68571350499056331620649629374, −0.78819259466702459849253676515, 0.78819259466702459849253676515, 1.68571350499056331620649629374, 2.27941211470107522943892843184, 3.71720204573071596317679520498, 4.22643559376834830035243990149, 5.48475911813519175659179726696, 5.92702329900097779158484049030, 6.40921079103539980944260610478, 7.33133297209796264462224475836, 7.83676740257569031736813294144

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