Properties

Label 2-7098-1.1-c1-0-13
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 − 1.63·5-s − 6-s + 7-s + 8-s + 9-s − 1.63·10-s − 6.10·11-s − 12-s + 14-s + 1.63·15-s + 16-s − 3.54·17-s + 18-s − 6.98·19-s − 1.63·20-s − 21-s − 6.10·22-s + 5.07·23-s − 24-s − 2.32·25-s − 27-s + 28-s − 7.10·29-s + 1.63·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.731·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.517·10-s − 1.84·11-s − 0.288·12-s + 0.267·14-s + 0.422·15-s + 0.250·16-s − 0.860·17-s + 0.235·18-s − 1.60·19-s − 0.365·20-s − 0.218·21-s − 1.30·22-s + 1.05·23-s − 0.204·24-s − 0.465·25-s − 0.192·27-s + 0.188·28-s − 1.31·29-s + 0.298·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.392510865\)
\(L(\frac12)\) \(\approx\) \(1.392510865\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.63T + 5T^{2} \)
11 \( 1 + 6.10T + 11T^{2} \)
17 \( 1 + 3.54T + 17T^{2} \)
19 \( 1 + 6.98T + 19T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 + 7.10T + 29T^{2} \)
31 \( 1 - 5.14T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + 4.12T + 43T^{2} \)
47 \( 1 - 12.5T + 47T^{2} \)
53 \( 1 + 13.9T + 53T^{2} \)
59 \( 1 - 1.33T + 59T^{2} \)
61 \( 1 - 9.77T + 61T^{2} \)
67 \( 1 - 9.30T + 67T^{2} \)
71 \( 1 + 6.97T + 71T^{2} \)
73 \( 1 + 5.42T + 73T^{2} \)
79 \( 1 - 0.425T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 - 5.43T + 89T^{2} \)
97 \( 1 - 9.00T + 97T^{2} \)
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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

−7.78350134179147322361823760741, −7.27305743596340259640261012114, −6.38434238990300634959892107166, −5.77568931198585271944228157830, −4.95125768193290168100736898344, −4.51161229496770260493688676114, −3.79461614844392518123725533236, −2.68010194846869781894402706162, −2.07846929920352107143716410788, −0.52373555138716405879647323749, 0.52373555138716405879647323749, 2.07846929920352107143716410788, 2.68010194846869781894402706162, 3.79461614844392518123725533236, 4.51161229496770260493688676114, 4.95125768193290168100736898344, 5.77568931198585271944228157830, 6.38434238990300634959892107166, 7.27305743596340259640261012114, 7.78350134179147322361823760741

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