Properties

Label 2-7098-1.1-c1-0-120
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 + 0.732·5-s − 6-s − 7-s + 8-s + 9-s + 0.732·10-s − 3.73·11-s − 12-s − 14-s − 0.732·15-s + 16-s + 0.267·17-s + 18-s + 4.46·19-s + 0.732·20-s + 21-s − 3.73·22-s + 3.46·23-s − 24-s − 4.46·25-s − 27-s − 28-s − 3·29-s − 0.732·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.327·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.231·10-s − 1.12·11-s − 0.288·12-s − 0.267·14-s − 0.189·15-s + 0.250·16-s + 0.0649·17-s + 0.235·18-s + 1.02·19-s + 0.163·20-s + 0.218·21-s − 0.795·22-s + 0.722·23-s − 0.204·24-s − 0.892·25-s − 0.192·27-s − 0.188·28-s − 0.557·29-s − 0.133·30-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 \( 1 - 0.732T + 5T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
17 \( 1 - 0.267T + 17T^{2} \)
19 \( 1 - 4.46T + 19T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 7.66T + 31T^{2} \)
37 \( 1 - 1.26T + 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 0.732T + 43T^{2} \)
47 \( 1 + 4.46T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 0.928T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 5.66T + 83T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 - 0.732T + 97T^{2} \)
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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

−7.58145509631131023494676008189, −6.73199870497150311580813820600, −6.01550537557317169249446694808, −5.39864750948483400149937065416, −4.99880740223038422986682391697, −3.98807138854505203231244587961, −3.21240106183492532178608480583, −2.41179960947626628687517788851, −1.38733439384469632512183222708, 0, 1.38733439384469632512183222708, 2.41179960947626628687517788851, 3.21240106183492532178608480583, 3.98807138854505203231244587961, 4.99880740223038422986682391697, 5.39864750948483400149937065416, 6.01550537557317169249446694808, 6.73199870497150311580813820600, 7.58145509631131023494676008189

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