Properties

Label 2-7098-1.1-c1-0-148
Degree $2$
Conductor $7098$
Sign $-1$
Analytic cond. $56.6778$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 3·11-s + 12-s + 14-s − 15-s + 16-s − 7·17-s + 18-s + 3·19-s − 20-s + 21-s − 3·22-s − 23-s + 24-s − 4·25-s + 27-s + 28-s − 29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 0.688·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s − 0.185·29-s − 0.182·30-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: yes
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 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.68083071479904466668655884042, −6.99032129461715610251582674174, −6.10261925745738300155527837065, −5.37599028452808911351827885632, −4.56585270381988719075615563165, −4.06251562111626197494276178200, −3.17893309422054665347416062595, −2.41731842051555287735923073505, −1.64739902798234594221402070321, 0, 1.64739902798234594221402070321, 2.41731842051555287735923073505, 3.17893309422054665347416062595, 4.06251562111626197494276178200, 4.56585270381988719075615563165, 5.37599028452808911351827885632, 6.10261925745738300155527837065, 6.99032129461715610251582674174, 7.68083071479904466668655884042

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