Properties

Label 2-7098-1.1-c1-0-124
Degree $2$
Conductor $7098$
Sign $-1$
Analytic cond. $56.6778$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 3·11-s + 12-s + 14-s + 15-s + 16-s − 7·17-s − 18-s − 3·19-s + 20-s − 21-s − 3·22-s − 23-s − 24-s − 4·25-s + 27-s − 28-s − 29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s − 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s − 0.185·29-s − 0.182·30-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: yes
Arithmetic: yes
Character: $\chi_{7098} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.72581737177485822420188579874, −6.75175741493069291678185912276, −6.52249189784401627590122093984, −5.73080737093857542602318321093, −4.50455671165503009783660844867, −3.97267520766741843024540343382, −2.89505913536008278898347549130, −2.18570989540565867051258059897, −1.40443853582166913560456681718, 0, 1.40443853582166913560456681718, 2.18570989540565867051258059897, 2.89505913536008278898347549130, 3.97267520766741843024540343382, 4.50455671165503009783660844867, 5.73080737093857542602318321093, 6.52249189784401627590122093984, 6.75175741493069291678185912276, 7.72581737177485822420188579874

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