Properties

Label 2-7098-1.1-c1-0-7
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 − 0.554·5-s + 6-s + 7-s − 8-s + 9-s + 0.554·10-s + 0.356·11-s − 12-s − 14-s + 0.554·15-s + 16-s − 3.80·17-s − 18-s − 5.74·19-s − 0.554·20-s − 21-s − 0.356·22-s − 2.44·23-s + 24-s − 4.69·25-s − 27-s + 28-s + 0.405·29-s − 0.554·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.248·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.175·10-s + 0.107·11-s − 0.288·12-s − 0.267·14-s + 0.143·15-s + 0.250·16-s − 0.922·17-s − 0.235·18-s − 1.31·19-s − 0.124·20-s − 0.218·21-s − 0.0760·22-s − 0.509·23-s + 0.204·24-s − 0.938·25-s − 0.192·27-s + 0.188·28-s + 0.0753·29-s − 0.101·30-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

\(L(1)\) \(\approx\) \(0.7002460633\)
\(L(\frac12)\) \(\approx\) \(0.7002460633\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.554T + 5T^{2} \)
11 \( 1 - 0.356T + 11T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 + 5.74T + 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 0.405T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 + 3.54T + 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 + 6.04T + 43T^{2} \)
47 \( 1 + 0.603T + 47T^{2} \)
53 \( 1 + 1.35T + 53T^{2} \)
59 \( 1 + 5.54T + 59T^{2} \)
61 \( 1 - 4.38T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 7.42T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 1.76T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 4.80T + 89T^{2} \)
97 \( 1 + 2.65T + 97T^{2} \)
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

−8.085661365533225188939820151009, −7.25559270490540527029359766238, −6.51850790934860162605601869789, −6.10895790391267112897609575707, −5.11188964812853834471134234842, −4.38468957820710088999350848869, −3.66011137781667334840890740850, −2.39167573010757554282269512651, −1.73108018270813567346492154628, −0.47883527559379491527470421676, 0.47883527559379491527470421676, 1.73108018270813567346492154628, 2.39167573010757554282269512651, 3.66011137781667334840890740850, 4.38468957820710088999350848869, 5.11188964812853834471134234842, 6.10895790391267112897609575707, 6.51850790934860162605601869789, 7.25559270490540527029359766238, 8.085661365533225188939820151009

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