Properties

Label 2-7098-1.1-c1-0-137
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 1.83·5-s − 6-s − 7-s − 8-s + 9-s − 1.83·10-s − 0.198·11-s + 12-s + 14-s + 1.83·15-s + 16-s + 0.445·17-s − 18-s − 1.57·19-s + 1.83·20-s − 21-s + 0.198·22-s + 9.19·23-s − 24-s − 1.63·25-s + 27-s − 28-s − 9.88·29-s − 1.83·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.820·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.579·10-s − 0.0597·11-s + 0.288·12-s + 0.267·14-s + 0.473·15-s + 0.250·16-s + 0.107·17-s − 0.235·18-s − 0.360·19-s + 0.410·20-s − 0.218·21-s + 0.0422·22-s + 1.91·23-s − 0.204·24-s − 0.327·25-s + 0.192·27-s − 0.188·28-s − 1.83·29-s − 0.334·30-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: \(1\)
Selberg data: \((2,\ 7098,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.83T + 5T^{2} \)
11 \( 1 + 0.198T + 11T^{2} \)
17 \( 1 - 0.445T + 17T^{2} \)
19 \( 1 + 1.57T + 19T^{2} \)
23 \( 1 - 9.19T + 23T^{2} \)
29 \( 1 + 9.88T + 29T^{2} \)
31 \( 1 + 6.88T + 31T^{2} \)
37 \( 1 + 5.13T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 - 8.51T + 47T^{2} \)
53 \( 1 + 1.38T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 0.307T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 7.90T + 73T^{2} \)
79 \( 1 - 6.98T + 79T^{2} \)
83 \( 1 + 3.19T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 + 4.95T + 97T^{2} \)
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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

−7.49068403065618155563513137887, −7.12078469304319977044284055681, −6.31841410640601667932074924635, −5.57587553284142770141569394303, −4.87243497855511456894876790140, −3.61828844920933141606611597551, −3.09603950672117069465681999398, −2.04193524717527508343077136088, −1.50532166815047045812582980333, 0, 1.50532166815047045812582980333, 2.04193524717527508343077136088, 3.09603950672117069465681999398, 3.61828844920933141606611597551, 4.87243497855511456894876790140, 5.57587553284142770141569394303, 6.31841410640601667932074924635, 7.12078469304319977044284055681, 7.49068403065618155563513137887

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