Properties

Label 2-7098-1.1-c1-0-9
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 − 4·5-s − 6-s − 7-s + 8-s + 9-s − 4·10-s + 3·11-s − 12-s − 14-s + 4·15-s + 16-s − 5·17-s + 18-s − 3·19-s − 4·20-s + 21-s + 3·22-s + 6·23-s − 24-s + 11·25-s − 27-s − 28-s − 9·29-s + 4·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.904·11-s − 0.288·12-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s − 0.688·19-s − 0.894·20-s + 0.218·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s − 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.730·30-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

\(L(1)\) \(\approx\) \(1.175691059\)
\(L(\frac12)\) \(\approx\) \(1.175691059\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−7.63669406487462342812298138630, −7.10788331147698024899342465561, −6.60672105014379586705386466380, −5.89854567188710674430052998224, −4.72374249512134126085632723418, −4.49202602268605232563722457210, −3.65591456157968936170908902374, −3.14105420626998936221520981013, −1.80967806877607615106627937729, −0.50033386573312125775273688836, 0.50033386573312125775273688836, 1.80967806877607615106627937729, 3.14105420626998936221520981013, 3.65591456157968936170908902374, 4.49202602268605232563722457210, 4.72374249512134126085632723418, 5.89854567188710674430052998224, 6.60672105014379586705386466380, 7.10788331147698024899342465561, 7.63669406487462342812298138630

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