Properties

Label 2-7098-1.1-c1-0-101
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 + 3.10·5-s + 6-s + 7-s + 8-s + 9-s + 3.10·10-s + 1.55·11-s + 12-s + 14-s + 3.10·15-s + 16-s − 1.24·17-s + 18-s − 4.73·19-s + 3.10·20-s + 21-s + 1.55·22-s + 5.10·23-s + 24-s + 4.66·25-s + 27-s + 28-s − 1.05·29-s + 3.10·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.39·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.983·10-s + 0.468·11-s + 0.288·12-s + 0.267·14-s + 0.802·15-s + 0.250·16-s − 0.302·17-s + 0.235·18-s − 1.08·19-s + 0.695·20-s + 0.218·21-s + 0.331·22-s + 1.06·23-s + 0.204·24-s + 0.932·25-s + 0.192·27-s + 0.188·28-s − 0.196·29-s + 0.567·30-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

\(L(1)\) \(\approx\) \(6.053546632\)
\(L(\frac12)\) \(\approx\) \(6.053546632\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.10T + 5T^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
17 \( 1 + 1.24T + 17T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 + 1.05T + 29T^{2} \)
31 \( 1 + 2.46T + 31T^{2} \)
37 \( 1 - 7.92T + 37T^{2} \)
41 \( 1 - 2.71T + 41T^{2} \)
43 \( 1 + 2.63T + 43T^{2} \)
47 \( 1 - 5.06T + 47T^{2} \)
53 \( 1 + 7.09T + 53T^{2} \)
59 \( 1 + 3.41T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 5.31T + 71T^{2} \)
73 \( 1 - 9.45T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 4.40T + 83T^{2} \)
89 \( 1 - 5.76T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.892495600161855759638537968807, −7.06268255430073328110318853215, −6.41029899867197567243277263481, −5.87688959022457688382903699798, −5.05132280527931610874781366254, −4.41651638248857286797078542450, −3.57335196784353281113865458532, −2.57521706810151933875405724276, −2.07001402417403767085478932875, −1.19392505837996915605495070702, 1.19392505837996915605495070702, 2.07001402417403767085478932875, 2.57521706810151933875405724276, 3.57335196784353281113865458532, 4.41651638248857286797078542450, 5.05132280527931610874781366254, 5.87688959022457688382903699798, 6.41029899867197567243277263481, 7.06268255430073328110318853215, 7.892495600161855759638537968807

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