Properties

Label 2-7098-1.1-c1-0-28
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 + 1.35·5-s − 6-s − 7-s + 8-s + 9-s + 1.35·10-s − 4.29·11-s − 12-s − 14-s − 1.35·15-s + 16-s + 2.66·17-s + 18-s − 4.29·19-s + 1.35·20-s + 21-s − 4.29·22-s − 4.63·23-s − 24-s − 3.15·25-s − 27-s − 28-s + 5.54·29-s − 1.35·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.606·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.429·10-s − 1.29·11-s − 0.288·12-s − 0.267·14-s − 0.350·15-s + 0.250·16-s + 0.646·17-s + 0.235·18-s − 0.985·19-s + 0.303·20-s + 0.218·21-s − 0.915·22-s − 0.965·23-s − 0.204·24-s − 0.631·25-s − 0.192·27-s − 0.188·28-s + 1.02·29-s − 0.247·30-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

\(L(1)\) \(\approx\) \(2.400631179\)
\(L(\frac12)\) \(\approx\) \(2.400631179\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.35T + 5T^{2} \)
11 \( 1 + 4.29T + 11T^{2} \)
17 \( 1 - 2.66T + 17T^{2} \)
19 \( 1 + 4.29T + 19T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 - 5.54T + 29T^{2} \)
31 \( 1 - 5.51T + 31T^{2} \)
37 \( 1 - 3.53T + 37T^{2} \)
41 \( 1 - 9.45T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 8.20T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 1.75T + 59T^{2} \)
61 \( 1 + 2.20T + 61T^{2} \)
67 \( 1 + 4.87T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 16.0T + 79T^{2} \)
83 \( 1 + 5.62T + 83T^{2} \)
89 \( 1 + 9.81T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.88730280475831362304196155958, −7.04543724780516734833404379745, −6.20834011992144226178335123324, −5.91868047272880425165128077794, −5.16264770486478922947440877943, −4.49454858810118512447501815079, −3.66938354277451455100568008582, −2.63412286386940625106858155014, −2.08178636012757952688851218408, −0.70419552267290399117817170350, 0.70419552267290399117817170350, 2.08178636012757952688851218408, 2.63412286386940625106858155014, 3.66938354277451455100568008582, 4.49454858810118512447501815079, 5.16264770486478922947440877943, 5.91868047272880425165128077794, 6.20834011992144226178335123324, 7.04543724780516734833404379745, 7.88730280475831362304196155958

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