Properties

Label 2-7098-1.1-c1-0-22
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 $0$

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 − 5·11-s + 12-s + 14-s + 15-s + 16-s − 3·17-s − 18-s + 19-s + 20-s − 21-s + 5·22-s + 3·23-s − 24-s − 4·25-s + 27-s − 28-s + 9·29-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 − 1.50·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.218·21-s + 1.06·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s + 1.67·29-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: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.481919160\)
\(L(\frac12)\) \(\approx\) \(1.481919160\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.88423413841871506203636604586, −7.53102487461663975706817859964, −6.61270975441273816602997295472, −6.01853701822204108385590735367, −5.14408356509184259794282746045, −4.37078468584442969243737745111, −3.19242539500705319758808535486, −2.63163126271274006390959216123, −1.91688586890288739737385366338, −0.64494925744861723051140272143, 0.64494925744861723051140272143, 1.91688586890288739737385366338, 2.63163126271274006390959216123, 3.19242539500705319758808535486, 4.37078468584442969243737745111, 5.14408356509184259794282746045, 6.01853701822204108385590735367, 6.61270975441273816602997295472, 7.53102487461663975706817859964, 7.88423413841871506203636604586

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