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Properties

Label 2-70699-1.1-c1-0-0
Degree $2$
Conductor $70699$
Sign $1$
Analytic cond. $564.534$
Root an. cond. $23.7599$
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  − 2·3-s − 2·4-s + 3·5-s + 7-s + 9-s − 3·11-s + 4·12-s − 4·13-s − 6·15-s + 4·16-s + 3·17-s + 19-s − 6·20-s − 2·21-s + 4·25-s + 4·27-s − 2·28-s − 6·29-s + 4·31-s + 6·33-s + 3·35-s − 2·36-s − 2·37-s + 8·39-s − 6·41-s + 43-s + 6·44-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.15·12-s − 1.10·13-s − 1.54·15-s + 16-s + 0.727·17-s + 0.229·19-s − 1.34·20-s − 0.436·21-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 0.152·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70699 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70699 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70699\)    =    \(19 \cdot 61^{2}\)
Sign: $1$
Analytic conductor: \(564.534\)
Root analytic conductor: \(23.7599\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70699,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5222981812\)
\(L(\frac12)\) \(\approx\) \(0.5222981812\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 - T \)
61 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−14.17766796771128, −13.52420768049186, −13.10833480765911, −12.72556903999067, −12.14009015819090, −11.71156794593411, −11.05394790802139, −10.44110621371669, −10.11783200183009, −9.667192995083579, −9.295995305941907, −8.504228912584119, −7.992425787705431, −7.459471155290003, −6.708009540079391, −6.125732385840841, −5.491813984138079, −5.311728909909750, −4.900845339946496, −4.320893782916487, −3.295593103055272, −2.733754145275043, −1.856826322624229, −1.245626990295787, −0.2739426667581370, 0.2739426667581370, 1.245626990295787, 1.856826322624229, 2.733754145275043, 3.295593103055272, 4.320893782916487, 4.900845339946496, 5.311728909909750, 5.491813984138079, 6.125732385840841, 6.708009540079391, 7.459471155290003, 7.992425787705431, 8.504228912584119, 9.295995305941907, 9.667192995083579, 10.11783200183009, 10.44110621371669, 11.05394790802139, 11.71156794593411, 12.14009015819090, 12.72556903999067, 13.10833480765911, 13.52420768049186, 14.17766796771128

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