Properties

Label 2-7098-1.1-c1-0-55
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 − 1.83·5-s + 6-s + 7-s + 8-s + 9-s − 1.83·10-s + 0.198·11-s + 12-s + 14-s − 1.83·15-s + 16-s + 0.445·17-s + 18-s + 1.57·19-s − 1.83·20-s + 21-s + 0.198·22-s + 9.19·23-s + 24-s − 1.63·25-s + 27-s + 28-s − 9.88·29-s − 1.83·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.820·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.579·10-s + 0.0597·11-s + 0.288·12-s + 0.267·14-s − 0.473·15-s + 0.250·16-s + 0.107·17-s + 0.235·18-s + 0.360·19-s − 0.410·20-s + 0.218·21-s + 0.0422·22-s + 1.91·23-s + 0.204·24-s − 0.327·25-s + 0.192·27-s + 0.188·28-s − 1.83·29-s − 0.334·30-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

\(L(1)\) \(\approx\) \(3.885865797\)
\(L(\frac12)\) \(\approx\) \(3.885865797\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.83T + 5T^{2} \)
11 \( 1 - 0.198T + 11T^{2} \)
17 \( 1 - 0.445T + 17T^{2} \)
19 \( 1 - 1.57T + 19T^{2} \)
23 \( 1 - 9.19T + 23T^{2} \)
29 \( 1 + 9.88T + 29T^{2} \)
31 \( 1 - 6.88T + 31T^{2} \)
37 \( 1 - 5.13T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 + 8.51T + 47T^{2} \)
53 \( 1 + 1.38T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 0.307T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 7.90T + 73T^{2} \)
79 \( 1 - 6.98T + 79T^{2} \)
83 \( 1 - 3.19T + 83T^{2} \)
89 \( 1 + 5.55T + 89T^{2} \)
97 \( 1 - 4.95T + 97T^{2} \)
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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

−7.79261572200143451696934839527, −7.35222552745934501397661780352, −6.61830566302479171472209124154, −5.72191079881529229714211653126, −4.91716620149217170617591458072, −4.33137186248650870160326361102, −3.54464616770944056319722973224, −2.96924414613553573216690738465, −1.99884975386622484648327850973, −0.896666456005942111763670546796, 0.896666456005942111763670546796, 1.99884975386622484648327850973, 2.96924414613553573216690738465, 3.54464616770944056319722973224, 4.33137186248650870160326361102, 4.91716620149217170617591458072, 5.72191079881529229714211653126, 6.61830566302479171472209124154, 7.35222552745934501397661780352, 7.79261572200143451696934839527

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