Properties

Label 2-7098-1.1-c1-0-58
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 4.04·5-s − 6-s − 7-s + 8-s + 9-s + 4.04·10-s − 5.74·11-s − 12-s − 14-s − 4.04·15-s + 16-s + 1.40·17-s + 18-s + 6.26·19-s + 4.04·20-s + 21-s − 5.74·22-s + 1.86·23-s − 24-s + 11.3·25-s − 27-s − 28-s − 3.20·29-s − 4.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.81·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.28·10-s − 1.73·11-s − 0.288·12-s − 0.267·14-s − 1.04·15-s + 0.250·16-s + 0.340·17-s + 0.235·18-s + 1.43·19-s + 0.905·20-s + 0.218·21-s − 1.22·22-s + 0.388·23-s − 0.204·24-s + 2.27·25-s − 0.192·27-s − 0.188·28-s − 0.595·29-s − 0.739·30-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

\(L(1)\) \(\approx\) \(3.633658663\)
\(L(\frac12)\) \(\approx\) \(3.633658663\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.04T + 5T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 - 6.26T + 19T^{2} \)
23 \( 1 - 1.86T + 23T^{2} \)
29 \( 1 + 3.20T + 29T^{2} \)
31 \( 1 - 8.63T + 31T^{2} \)
37 \( 1 - 3.36T + 37T^{2} \)
41 \( 1 + 2.59T + 41T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + 8.27T + 47T^{2} \)
53 \( 1 + 3.16T + 53T^{2} \)
59 \( 1 - 7.50T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 9.20T + 67T^{2} \)
71 \( 1 + 16.8T + 71T^{2} \)
73 \( 1 + 0.317T + 73T^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 - 7.10T + 83T^{2} \)
89 \( 1 + 8.02T + 89T^{2} \)
97 \( 1 - 18.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.72167511487973625100818866840, −7.00301308762768441448181107781, −6.26703008594601874714569004943, −5.75933181280254690549751143396, −5.09327453961208513195758704707, −4.89399667989101601478955276663, −3.36922615686282332945954399746, −2.73959738439111285985215294054, −1.98225076921600848391361714243, −0.907321693321209489941544229100, 0.907321693321209489941544229100, 1.98225076921600848391361714243, 2.73959738439111285985215294054, 3.36922615686282332945954399746, 4.89399667989101601478955276663, 5.09327453961208513195758704707, 5.75933181280254690549751143396, 6.26703008594601874714569004943, 7.00301308762768441448181107781, 7.72167511487973625100818866840

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