Properties

Label 2-7098-1.1-c1-0-11
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 − 2.24·5-s + 6-s + 7-s − 8-s + 9-s + 2.24·10-s + 0.692·11-s − 12-s − 14-s + 2.24·15-s + 16-s − 2.44·17-s − 18-s + 3.40·19-s − 2.24·20-s − 21-s − 0.692·22-s − 0.753·23-s + 24-s + 0.0489·25-s − 27-s + 28-s − 3.66·29-s − 2.24·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.710·10-s + 0.208·11-s − 0.288·12-s − 0.267·14-s + 0.580·15-s + 0.250·16-s − 0.593·17-s − 0.235·18-s + 0.781·19-s − 0.502·20-s − 0.218·21-s − 0.147·22-s − 0.157·23-s + 0.204·24-s + 0.00978·25-s − 0.192·27-s + 0.188·28-s − 0.680·29-s − 0.410·30-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

\(L(1)\) \(\approx\) \(0.7135811634\)
\(L(\frac12)\) \(\approx\) \(0.7135811634\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.24T + 5T^{2} \)
11 \( 1 - 0.692T + 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + 0.753T + 23T^{2} \)
29 \( 1 + 3.66T + 29T^{2} \)
31 \( 1 + 4.40T + 31T^{2} \)
37 \( 1 - 6.96T + 37T^{2} \)
41 \( 1 - 5.63T + 41T^{2} \)
43 \( 1 + 1.64T + 43T^{2} \)
47 \( 1 - 2.10T + 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 - 4.96T + 59T^{2} \)
61 \( 1 + 5.09T + 61T^{2} \)
67 \( 1 - 7.84T + 67T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 1.97T + 79T^{2} \)
83 \( 1 + 9.22T + 83T^{2} \)
89 \( 1 + 3.44T + 89T^{2} \)
97 \( 1 - 4.46T + 97T^{2} \)
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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

−7.80270686593654787236625370362, −7.43790631089011507672821448585, −6.71036929227936872637472162304, −5.90604861422710567767516032087, −5.18301064037212026806011823830, −4.26179456586535489128579673293, −3.69372055071814788785623113185, −2.58938766105828621995092389567, −1.53563747636027874423913510635, −0.50923830242280401073080917065, 0.50923830242280401073080917065, 1.53563747636027874423913510635, 2.58938766105828621995092389567, 3.69372055071814788785623113185, 4.26179456586535489128579673293, 5.18301064037212026806011823830, 5.90604861422710567767516032087, 6.71036929227936872637472162304, 7.43790631089011507672821448585, 7.80270686593654787236625370362

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