Properties

Label 2-7098-1.1-c1-0-103
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 0.692·5-s + 6-s + 7-s − 8-s + 9-s − 0.692·10-s + 0.664·11-s − 12-s − 14-s − 0.692·15-s + 16-s − 7.74·17-s − 18-s + 3.89·19-s + 0.692·20-s − 21-s − 0.664·22-s + 0.841·23-s + 24-s − 4.52·25-s − 27-s + 28-s + 8.98·29-s + 0.692·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.309·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.218·10-s + 0.200·11-s − 0.288·12-s − 0.267·14-s − 0.178·15-s + 0.250·16-s − 1.87·17-s − 0.235·18-s + 0.894·19-s + 0.154·20-s − 0.218·21-s − 0.141·22-s + 0.175·23-s + 0.204·24-s − 0.904·25-s − 0.192·27-s + 0.188·28-s + 1.66·29-s + 0.126·30-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: \(1\)
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 \( 1 - 0.692T + 5T^{2} \)
11 \( 1 - 0.664T + 11T^{2} \)
17 \( 1 + 7.74T + 17T^{2} \)
19 \( 1 - 3.89T + 19T^{2} \)
23 \( 1 - 0.841T + 23T^{2} \)
29 \( 1 - 8.98T + 29T^{2} \)
31 \( 1 + 6.26T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 - 9.87T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 6.05T + 61T^{2} \)
67 \( 1 - 2.98T + 67T^{2} \)
71 \( 1 + 9.93T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 1.60T + 79T^{2} \)
83 \( 1 + 4.39T + 83T^{2} \)
89 \( 1 + 0.454T + 89T^{2} \)
97 \( 1 + 8.61T + 97T^{2} \)
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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

−7.41398168829002998366486834258, −7.08857840997328315886271952872, −6.14097658052542788362709031694, −5.73114904267701630817968365503, −4.72212620838503450351031735674, −4.13647589826749507063462039709, −2.92096599066984843548267945902, −2.05059746865616231182288180898, −1.19906902808632053387626735729, 0, 1.19906902808632053387626735729, 2.05059746865616231182288180898, 2.92096599066984843548267945902, 4.13647589826749507063462039709, 4.72212620838503450351031735674, 5.73114904267701630817968365503, 6.14097658052542788362709031694, 7.08857840997328315886271952872, 7.41398168829002998366486834258

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