Properties

Label 2-7098-1.1-c1-0-53
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 + 0.332·5-s − 6-s + 7-s + 8-s + 9-s + 0.332·10-s + 2.61·11-s − 12-s + 14-s − 0.332·15-s + 16-s − 3.88·17-s + 18-s + 5.61·19-s + 0.332·20-s − 21-s + 2.61·22-s + 4.21·23-s − 24-s − 4.88·25-s − 27-s + 28-s − 1.18·29-s − 0.332·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.148·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.105·10-s + 0.787·11-s − 0.288·12-s + 0.267·14-s − 0.0859·15-s + 0.250·16-s − 0.942·17-s + 0.235·18-s + 1.28·19-s + 0.0744·20-s − 0.218·21-s + 0.556·22-s + 0.878·23-s − 0.204·24-s − 0.977·25-s − 0.192·27-s + 0.188·28-s − 0.220·29-s − 0.0607·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.218413961\)
\(L(\frac12)\) \(\approx\) \(3.218413961\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.332T + 5T^{2} \)
11 \( 1 - 2.61T + 11T^{2} \)
17 \( 1 + 3.88T + 17T^{2} \)
19 \( 1 - 5.61T + 19T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 + 1.18T + 29T^{2} \)
31 \( 1 - 7.07T + 31T^{2} \)
37 \( 1 - 0.576T + 37T^{2} \)
41 \( 1 + 0.521T + 41T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 9.71T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 7.43T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 0.944T + 71T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 + 0.943T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + 5.81T + 97T^{2} \)
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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

−7.63217229873804130247876182966, −7.17198677708316561533131304193, −6.31415720364110047035034433983, −5.89631668851110294990092136872, −5.00771899207757915844734941167, −4.50531741288826350554888993158, −3.72300671011040474992771122526, −2.80639291034982433563033976854, −1.80575632725015360693985906986, −0.882474710391900696021847199186, 0.882474710391900696021847199186, 1.80575632725015360693985906986, 2.80639291034982433563033976854, 3.72300671011040474992771122526, 4.50531741288826350554888993158, 5.00771899207757915844734941167, 5.89631668851110294990092136872, 6.31415720364110047035034433983, 7.17198677708316561533131304193, 7.63217229873804130247876182966

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