Properties

Label 2-7098-1.1-c1-0-67
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 + 1.69·5-s − 6-s − 7-s + 8-s + 9-s + 1.69·10-s + 3.15·11-s − 12-s − 14-s − 1.69·15-s + 16-s + 7.74·17-s + 18-s + 3.15·19-s + 1.69·20-s + 21-s + 3.15·22-s + 7.89·23-s − 24-s − 2.13·25-s − 27-s − 28-s − 4.96·29-s − 1.69·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.756·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.535·10-s + 0.952·11-s − 0.288·12-s − 0.267·14-s − 0.436·15-s + 0.250·16-s + 1.87·17-s + 0.235·18-s + 0.724·19-s + 0.378·20-s + 0.218·21-s + 0.673·22-s + 1.64·23-s − 0.204·24-s − 0.427·25-s − 0.192·27-s − 0.188·28-s − 0.921·29-s − 0.308·30-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

\(L(1)\) \(\approx\) \(3.706409094\)
\(L(\frac12)\) \(\approx\) \(3.706409094\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 - 3.15T + 11T^{2} \)
17 \( 1 - 7.74T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 - 7.89T + 23T^{2} \)
29 \( 1 + 4.96T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 + 0.185T + 37T^{2} \)
41 \( 1 - 0.978T + 41T^{2} \)
43 \( 1 + 8.19T + 43T^{2} \)
47 \( 1 - 2.78T + 47T^{2} \)
53 \( 1 + 0.652T + 53T^{2} \)
59 \( 1 + 4.80T + 59T^{2} \)
61 \( 1 - 3.21T + 61T^{2} \)
67 \( 1 + 4.52T + 67T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + 7.33T + 79T^{2} \)
83 \( 1 + 8.32T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 7.83T + 97T^{2} \)
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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

−7.58321243888235596129821135800, −7.12226695456431188278024949875, −6.31114603329833797337362154568, −5.77927732917681737026692716288, −5.27288193137734556227974655274, −4.47625231756729024719877778905, −3.48617716682372907270574098783, −2.99267196603154114743720235527, −1.69405882352552328190478774719, −0.990368982624148812032734833212, 0.990368982624148812032734833212, 1.69405882352552328190478774719, 2.99267196603154114743720235527, 3.48617716682372907270574098783, 4.47625231756729024719877778905, 5.27288193137734556227974655274, 5.77927732917681737026692716288, 6.31114603329833797337362154568, 7.12226695456431188278024949875, 7.58321243888235596129821135800

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