Properties

Label 2-7098-1.1-c1-0-144
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 0.246·5-s − 6-s + 7-s + 8-s + 9-s + 0.246·10-s + 5.40·11-s − 12-s + 14-s − 0.246·15-s + 16-s − 4.04·17-s + 18-s − 3.80·19-s + 0.246·20-s − 21-s + 5.40·22-s − 9.34·23-s − 24-s − 4.93·25-s − 27-s + 28-s − 3.26·29-s − 0.246·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.110·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.0781·10-s + 1.62·11-s − 0.288·12-s + 0.267·14-s − 0.0637·15-s + 0.250·16-s − 0.982·17-s + 0.235·18-s − 0.872·19-s + 0.0552·20-s − 0.218·21-s + 1.15·22-s − 1.94·23-s − 0.204·24-s − 0.987·25-s − 0.192·27-s + 0.188·28-s − 0.606·29-s − 0.0450·30-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: \(1\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.246T + 5T^{2} \)
11 \( 1 - 5.40T + 11T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 + 9.34T + 23T^{2} \)
29 \( 1 + 3.26T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 - 1.35T + 37T^{2} \)
41 \( 1 + 0.246T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 + 4.38T + 47T^{2} \)
53 \( 1 + 8.67T + 53T^{2} \)
59 \( 1 + 0.466T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 5.71T + 67T^{2} \)
71 \( 1 - 6.98T + 71T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
79 \( 1 + 5.75T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 17.9T + 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−7.42871895269471639790724283733, −6.53231168941198109931448455183, −6.24134687254243288913979807974, −5.58527297937232432189809376816, −4.47905128903564765895006370180, −4.23062313281700466384723449318, −3.43237550627539072188821035241, −2.03443820849303010942486625168, −1.62032934137852590389601881329, 0, 1.62032934137852590389601881329, 2.03443820849303010942486625168, 3.43237550627539072188821035241, 4.23062313281700466384723449318, 4.47905128903564765895006370180, 5.58527297937232432189809376816, 6.24134687254243288913979807974, 6.53231168941198109931448455183, 7.42871895269471639790724283733

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