Properties

Label 2-7098-1.1-c1-0-52
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 − 2.41·5-s + 6-s + 7-s + 8-s + 9-s − 2.41·10-s + 3.24·11-s + 12-s + 14-s − 2.41·15-s + 16-s + 1.80·17-s + 18-s − 2.73·19-s − 2.41·20-s + 21-s + 3.24·22-s − 6.58·23-s + 24-s + 0.832·25-s + 27-s + 28-s + 6.22·29-s − 2.41·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.08·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.979·11-s + 0.288·12-s + 0.267·14-s − 0.623·15-s + 0.250·16-s + 0.437·17-s + 0.235·18-s − 0.628·19-s − 0.540·20-s + 0.218·21-s + 0.692·22-s − 1.37·23-s + 0.204·24-s + 0.166·25-s + 0.192·27-s + 0.188·28-s + 1.15·29-s − 0.440·30-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

\(L(1)\) \(\approx\) \(3.761726962\)
\(L(\frac12)\) \(\approx\) \(3.761726962\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 5T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
17 \( 1 - 1.80T + 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + 6.58T + 23T^{2} \)
29 \( 1 - 6.22T + 29T^{2} \)
31 \( 1 - 2.72T + 31T^{2} \)
37 \( 1 - 8.93T + 37T^{2} \)
41 \( 1 - 9.15T + 41T^{2} \)
43 \( 1 + 3.95T + 43T^{2} \)
47 \( 1 - 9.77T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 4.50T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 + 2.54T + 67T^{2} \)
71 \( 1 + 1.50T + 71T^{2} \)
73 \( 1 + 0.786T + 73T^{2} \)
79 \( 1 + 2.41T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−7.82055953949008977692079660702, −7.42811044961945008797415373243, −6.39809700872875585934267916066, −6.00079425390356674788822760075, −4.74538642623872064068852892380, −4.23588211455717428837526343113, −3.77057729435417352303869476737, −2.89115902594235993024331458770, −2.00071980547430035999588849957, −0.881392460209941472974825371294, 0.881392460209941472974825371294, 2.00071980547430035999588849957, 2.89115902594235993024331458770, 3.77057729435417352303869476737, 4.23588211455717428837526343113, 4.74538642623872064068852892380, 6.00079425390356674788822760075, 6.39809700872875585934267916066, 7.42811044961945008797415373243, 7.82055953949008977692079660702

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