Properties

Label 2-7098-1.1-c1-0-30
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 − 1.78·5-s − 6-s + 7-s + 8-s + 9-s − 1.78·10-s − 3.17·11-s − 12-s + 14-s + 1.78·15-s + 16-s + 5.56·17-s + 18-s + 6.18·19-s − 1.78·20-s − 21-s − 3.17·22-s + 6.13·23-s − 24-s − 1.80·25-s − 27-s + 28-s + 2.07·29-s + 1.78·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.799·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.565·10-s − 0.957·11-s − 0.288·12-s + 0.267·14-s + 0.461·15-s + 0.250·16-s + 1.34·17-s + 0.235·18-s + 1.41·19-s − 0.399·20-s − 0.218·21-s − 0.676·22-s + 1.27·23-s − 0.204·24-s − 0.360·25-s − 0.192·27-s + 0.188·28-s + 0.385·29-s + 0.326·30-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

\(L(1)\) \(\approx\) \(2.280023145\)
\(L(\frac12)\) \(\approx\) \(2.280023145\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.78T + 5T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
19 \( 1 - 6.18T + 19T^{2} \)
23 \( 1 - 6.13T + 23T^{2} \)
29 \( 1 - 2.07T + 29T^{2} \)
31 \( 1 + 5.63T + 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
41 \( 1 + 1.49T + 41T^{2} \)
43 \( 1 + 9.63T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 3.60T + 53T^{2} \)
59 \( 1 + 2.78T + 59T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 0.598T + 71T^{2} \)
73 \( 1 - 0.423T + 73T^{2} \)
79 \( 1 - 6.96T + 79T^{2} \)
83 \( 1 - 4.30T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−7.63579330759077811236167242314, −7.39566124586540546753552323212, −6.47278868770403154187563858820, −5.53440335488009348815732211346, −5.15563585566830228544786140967, −4.56349956736145097315329734796, −3.41898515780946498671885636474, −3.14711217292492276841329924958, −1.77974891376860752562381094603, −0.72277682292045604824703090952, 0.72277682292045604824703090952, 1.77974891376860752562381094603, 3.14711217292492276841329924958, 3.41898515780946498671885636474, 4.56349956736145097315329734796, 5.15563585566830228544786140967, 5.53440335488009348815732211346, 6.47278868770403154187563858820, 7.39566124586540546753552323212, 7.63579330759077811236167242314

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