Properties

Label 2-7098-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 3·5-s + 6-s + 7-s − 8-s + 9-s + 3·10-s − 11-s − 12-s − 14-s + 3·15-s + 16-s + 7·17-s − 18-s − 19-s − 3·20-s − 21-s + 22-s − 7·23-s + 24-s + 4·25-s − 27-s + 28-s + 3·29-s − 3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.301·11-s − 0.288·12-s − 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s + 0.204·24-s + 4/5·25-s − 0.192·27-s + 0.188·28-s + 0.557·29-s − 0.547·30-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: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6690735170\)
\(L(\frac12)\) \(\approx\) \(0.6690735170\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.898076037134797332948446484117, −7.58663423389632535617995531918, −6.64452163354514621154107499610, −5.91794309631263039139114586698, −5.16541177539205916739574727700, −4.27520572783648654100867284139, −3.64227404025705936992976709258, −2.68602090632586806212409773589, −1.46853372510881575423801935182, −0.50330060895702483123303804703, 0.50330060895702483123303804703, 1.46853372510881575423801935182, 2.68602090632586806212409773589, 3.64227404025705936992976709258, 4.27520572783648654100867284139, 5.16541177539205916739574727700, 5.91794309631263039139114586698, 6.64452163354514621154107499610, 7.58663423389632535617995531918, 7.898076037134797332948446484117

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