Properties

Label 2-7098-1.1-c1-0-73
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 − 3.41·5-s − 6-s − 7-s − 8-s + 9-s + 3.41·10-s − 3.24·11-s + 12-s + 14-s − 3.41·15-s + 16-s + 1.80·17-s − 18-s − 1.93·19-s − 3.41·20-s − 21-s + 3.24·22-s + 3.28·23-s − 24-s + 6.66·25-s + 27-s − 28-s − 3.64·29-s + 3.41·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.52·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.07·10-s − 0.979·11-s + 0.288·12-s + 0.267·14-s − 0.881·15-s + 0.250·16-s + 0.437·17-s − 0.235·18-s − 0.444·19-s − 0.763·20-s − 0.218·21-s + 0.692·22-s + 0.684·23-s − 0.204·24-s + 1.33·25-s + 0.192·27-s − 0.188·28-s − 0.676·29-s + 0.623·30-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 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
17 \( 1 - 1.80T + 17T^{2} \)
19 \( 1 + 1.93T + 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 + 3.64T + 29T^{2} \)
31 \( 1 - 3.10T + 31T^{2} \)
37 \( 1 - 4.80T + 37T^{2} \)
41 \( 1 - 4.58T + 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 2.46T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 2.66T + 71T^{2} \)
73 \( 1 + 0.368T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 1.43T + 83T^{2} \)
89 \( 1 + 5.10T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−7.62934484036565802562892396678, −7.34727254815817448897412980366, −6.43080282619867355097342929159, −5.51862235460595579333180261737, −4.51242583228203543330532934938, −3.85558359062283771052041389534, −3.02624309303735036376971597378, −2.44620484750367778742196299500, −1.02631367090756010867941940444, 0, 1.02631367090756010867941940444, 2.44620484750367778742196299500, 3.02624309303735036376971597378, 3.85558359062283771052041389534, 4.51242583228203543330532934938, 5.51862235460595579333180261737, 6.43080282619867355097342929159, 7.34727254815817448897412980366, 7.62934484036565802562892396678

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