Properties

Label 2-7098-1.1-c1-0-93
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 + 2.83·5-s + 6-s + 7-s + 8-s + 9-s + 2.83·10-s + 0.198·11-s + 12-s + 14-s + 2.83·15-s + 16-s + 0.445·17-s + 18-s − 1.01·19-s + 2.83·20-s + 21-s + 0.198·22-s − 5.03·23-s + 24-s + 3.03·25-s + 27-s + 28-s + 4.34·29-s + 2.83·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.896·10-s + 0.0597·11-s + 0.288·12-s + 0.267·14-s + 0.731·15-s + 0.250·16-s + 0.107·17-s + 0.235·18-s − 0.233·19-s + 0.633·20-s + 0.218·21-s + 0.0422·22-s − 1.05·23-s + 0.204·24-s + 0.606·25-s + 0.192·27-s + 0.188·28-s + 0.806·29-s + 0.517·30-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

\(L(1)\) \(\approx\) \(5.812406498\)
\(L(\frac12)\) \(\approx\) \(5.812406498\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.83T + 5T^{2} \)
11 \( 1 - 0.198T + 11T^{2} \)
17 \( 1 - 0.445T + 17T^{2} \)
19 \( 1 + 1.01T + 19T^{2} \)
23 \( 1 + 5.03T + 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 - 2.21T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 + 0.443T + 41T^{2} \)
43 \( 1 - 7.64T + 43T^{2} \)
47 \( 1 - 3.63T + 47T^{2} \)
53 \( 1 - 6.50T + 53T^{2} \)
59 \( 1 - 5.45T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 - 5.80T + 71T^{2} \)
73 \( 1 + 0.641T + 73T^{2} \)
79 \( 1 - 1.67T + 79T^{2} \)
83 \( 1 - 6.42T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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

−8.017030762737069606137652934804, −7.01889652385426573935312669833, −6.52556257867114977629399441561, −5.66789716928235038946449469408, −5.25017497592586264527956242862, −4.28207833268321548217227294570, −3.64811352675982167306828202880, −2.50573613104622215107525494693, −2.15495236206106371409399263189, −1.14781709830070000110913214021, 1.14781709830070000110913214021, 2.15495236206106371409399263189, 2.50573613104622215107525494693, 3.64811352675982167306828202880, 4.28207833268321548217227294570, 5.25017497592586264527956242862, 5.66789716928235038946449469408, 6.52556257867114977629399441561, 7.01889652385426573935312669833, 8.017030762737069606137652934804

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