Properties

Label 2-7098-1.1-c1-0-63
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 + 3.56·5-s − 6-s + 7-s + 8-s + 9-s + 3.56·10-s − 5.56·11-s − 12-s + 14-s − 3.56·15-s + 16-s + 6.68·17-s + 18-s − 1.56·19-s + 3.56·20-s − 21-s − 5.56·22-s + 6.68·23-s − 24-s + 7.68·25-s − 27-s + 28-s + 1.56·29-s − 3.56·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.59·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s − 1.67·11-s − 0.288·12-s + 0.267·14-s − 0.919·15-s + 0.250·16-s + 1.62·17-s + 0.235·18-s − 0.358·19-s + 0.796·20-s − 0.218·21-s − 1.18·22-s + 1.39·23-s − 0.204·24-s + 1.53·25-s − 0.192·27-s + 0.188·28-s + 0.289·29-s − 0.650·30-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

\(L(1)\) \(\approx\) \(3.888707928\)
\(L(\frac12)\) \(\approx\) \(3.888707928\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.56T + 5T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
19 \( 1 + 1.56T + 19T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 6.43T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 4.87T + 53T^{2} \)
59 \( 1 + 4.24T + 59T^{2} \)
61 \( 1 + 1.56T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 - 9.36T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−7.59365591100043862340432917494, −7.22669544383508386576816102649, −6.22866384066165521370774853818, −5.56180413767529545714201956514, −5.33917014821716389344046471942, −4.74737368744550953168740472843, −3.49679758868648860682835699441, −2.64886199597523044245473640126, −1.95860051293225030554226611163, −0.956285702528353515601824138628, 0.956285702528353515601824138628, 1.95860051293225030554226611163, 2.64886199597523044245473640126, 3.49679758868648860682835699441, 4.74737368744550953168740472843, 5.33917014821716389344046471942, 5.56180413767529545714201956514, 6.22866384066165521370774853818, 7.22669544383508386576816102649, 7.59365591100043862340432917494

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