Properties

Label 2-7098-1.1-c1-0-57
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 + 0.554·5-s − 6-s + 7-s − 8-s + 9-s − 0.554·10-s + 1.64·11-s + 12-s − 14-s + 0.554·15-s + 16-s + 2.91·17-s − 18-s + 6.85·19-s + 0.554·20-s + 21-s − 1.64·22-s + 2.93·23-s − 24-s − 4.69·25-s + 27-s + 28-s − 3.69·29-s − 0.554·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.248·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.175·10-s + 0.495·11-s + 0.288·12-s − 0.267·14-s + 0.143·15-s + 0.250·16-s + 0.706·17-s − 0.235·18-s + 1.57·19-s + 0.124·20-s + 0.218·21-s − 0.350·22-s + 0.612·23-s − 0.204·24-s − 0.938·25-s + 0.192·27-s + 0.188·28-s − 0.685·29-s − 0.101·30-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

\(L(1)\) \(\approx\) \(2.357856273\)
\(L(\frac12)\) \(\approx\) \(2.357856273\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.554T + 5T^{2} \)
11 \( 1 - 1.64T + 11T^{2} \)
17 \( 1 - 2.91T + 17T^{2} \)
19 \( 1 - 6.85T + 19T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 + 3.69T + 29T^{2} \)
31 \( 1 + 1.13T + 31T^{2} \)
37 \( 1 - 7.93T + 37T^{2} \)
41 \( 1 - 2.55T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 5.13T + 53T^{2} \)
59 \( 1 + 0.158T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 1.69T + 67T^{2} \)
71 \( 1 - 1.01T + 71T^{2} \)
73 \( 1 - 5.97T + 73T^{2} \)
79 \( 1 - 8.92T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−7.892128159958206622624521732392, −7.46003293457766768464474506775, −6.80143590300971882607098489713, −5.80922771234748106969122571499, −5.29917181101515930402398739097, −4.18801995663651767370230393506, −3.41580840066717853193681593877, −2.61908993193749558177314158561, −1.66786997597042686232069740173, −0.904383643580367007923322534680, 0.904383643580367007923322534680, 1.66786997597042686232069740173, 2.61908993193749558177314158561, 3.41580840066717853193681593877, 4.18801995663651767370230393506, 5.29917181101515930402398739097, 5.80922771234748106969122571499, 6.80143590300971882607098489713, 7.46003293457766768464474506775, 7.892128159958206622624521732392

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