Properties

Label 2-7098-1.1-c1-0-85
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 + 6-s + 7-s + 8-s + 9-s + 3·11-s + 12-s + 14-s + 16-s + 3·17-s + 18-s + 5·19-s + 21-s + 3·22-s + 6·23-s + 24-s − 5·25-s + 27-s + 28-s − 3·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.14·19-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + 1/6·36-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.070156817\)
\(L(\frac12)\) \(\approx\) \(5.070156817\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−7.61033296333440848965148475886, −7.41050604158457718911977471618, −6.54649251029449046549842637035, −5.65816118444786466531912973433, −5.15332984627406569455164982731, −4.20078605597733533031771243492, −3.59633204347625063438465229282, −2.92908234127268278046358107803, −1.88189293402908322204857100573, −1.09105901653179902905242278073, 1.09105901653179902905242278073, 1.88189293402908322204857100573, 2.92908234127268278046358107803, 3.59633204347625063438465229282, 4.20078605597733533031771243492, 5.15332984627406569455164982731, 5.65816118444786466531912973433, 6.54649251029449046549842637035, 7.41050604158457718911977471618, 7.61033296333440848965148475886

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