Properties

Label 2-7098-1.1-c1-0-17
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 − 3.04·5-s − 6-s − 7-s + 8-s + 9-s − 3.04·10-s + 2.13·11-s − 12-s − 14-s + 3.04·15-s + 16-s − 1.40·17-s + 18-s + 2.13·19-s − 3.04·20-s + 21-s + 2.13·22-s − 2.26·23-s − 24-s + 4.29·25-s − 27-s − 28-s − 2.58·29-s + 3.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.36·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.964·10-s + 0.644·11-s − 0.288·12-s − 0.267·14-s + 0.787·15-s + 0.250·16-s − 0.340·17-s + 0.235·18-s + 0.490·19-s − 0.681·20-s + 0.218·21-s + 0.455·22-s − 0.473·23-s − 0.204·24-s + 0.859·25-s − 0.192·27-s − 0.188·28-s − 0.479·29-s + 0.556·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.502045373\)
\(L(\frac12)\) \(\approx\) \(1.502045373\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.04T + 5T^{2} \)
11 \( 1 - 2.13T + 11T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 - 2.13T + 19T^{2} \)
23 \( 1 + 2.26T + 23T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 4.43T + 41T^{2} \)
43 \( 1 + 9.55T + 43T^{2} \)
47 \( 1 + 3.98T + 47T^{2} \)
53 \( 1 - 6.46T + 53T^{2} \)
59 \( 1 + 3.44T + 59T^{2} \)
61 \( 1 - 9.98T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 4.21T + 71T^{2} \)
73 \( 1 - 5.84T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 8.94T + 83T^{2} \)
89 \( 1 + 3.73T + 89T^{2} \)
97 \( 1 + 5.10T + 97T^{2} \)
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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

−7.83836581535813764216806971181, −6.97571308322249683287413505120, −6.67737889969777734493315144302, −5.73214210635757272560599836299, −5.08897951785586896716068707039, −4.15662587948371313749634278917, −3.83757511907556622262080628487, −3.03652081732347157195545113311, −1.82253461326485157326253031006, −0.56881597555184533844271705534, 0.56881597555184533844271705534, 1.82253461326485157326253031006, 3.03652081732347157195545113311, 3.83757511907556622262080628487, 4.15662587948371313749634278917, 5.08897951785586896716068707039, 5.73214210635757272560599836299, 6.67737889969777734493315144302, 6.97571308322249683287413505120, 7.83836581535813764216806971181

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