Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 3.27·5-s − 6-s − 7-s + 8-s + 9-s − 3.27·10-s − 5.27·11-s − 12-s − 14-s + 3.27·15-s + 16-s − 7.27·17-s + 18-s − 5.27·19-s − 3.27·20-s + 21-s − 5.27·22-s − 5.27·23-s − 24-s + 5.72·25-s − 27-s − 28-s + 0.725·29-s + 3.27·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.46·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.03·10-s − 1.59·11-s − 0.288·12-s − 0.267·14-s + 0.845·15-s + 0.250·16-s − 1.76·17-s + 0.235·18-s − 1.21·19-s − 0.732·20-s + 0.218·21-s − 1.12·22-s − 1.09·23-s − 0.204·24-s + 1.14·25-s − 0.192·27-s − 0.188·28-s + 0.134·29-s + 0.597·30-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: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3090460083\)
\(L(\frac12)\) \(\approx\) \(0.3090460083\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
19 \( 1 + 5.27T + 19T^{2} \)
23 \( 1 + 5.27T + 23T^{2} \)
29 \( 1 - 0.725T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 3.27T + 37T^{2} \)
41 \( 1 - 8.54T + 41T^{2} \)
43 \( 1 - 5.27T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 4.72T + 61T^{2} \)
67 \( 1 - 2.54T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.82T + 73T^{2} \)
79 \( 1 + 2.54T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−7.66355750417991140697761149300, −7.29520075964807195407607261495, −6.45522411028721007232063303897, −5.81326612013499965031562730859, −4.98440869850844187168806825124, −4.22906947203251665901274850447, −3.93396769669934073899606104063, −2.79288000868637548437647632133, −2.07170418726121768412641619059, −0.24003321774572802507874822758, 0.24003321774572802507874822758, 2.07170418726121768412641619059, 2.79288000868637548437647632133, 3.93396769669934073899606104063, 4.22906947203251665901274850447, 4.98440869850844187168806825124, 5.81326612013499965031562730859, 6.45522411028721007232063303897, 7.29520075964807195407607261495, 7.66355750417991140697761149300

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