Properties

Label 2-619-1.1-c1-0-1
Degree $2$
Conductor $619$
Sign $1$
Analytic cond. $4.94273$
Root an. cond. $2.22322$
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  − 2.42·2-s − 3.02·3-s + 3.90·4-s − 0.0323·5-s + 7.35·6-s − 3.37·7-s − 4.62·8-s + 6.16·9-s + 0.0784·10-s + 0.333·11-s − 11.8·12-s − 1.65·13-s + 8.20·14-s + 0.0977·15-s + 3.42·16-s − 1.50·17-s − 14.9·18-s − 5.02·19-s − 0.126·20-s + 10.2·21-s − 0.809·22-s − 5.35·23-s + 13.9·24-s − 4.99·25-s + 4.02·26-s − 9.57·27-s − 13.1·28-s + ⋯
L(s)  = 1  − 1.71·2-s − 1.74·3-s + 1.95·4-s − 0.0144·5-s + 3.00·6-s − 1.27·7-s − 1.63·8-s + 2.05·9-s + 0.0248·10-s + 0.100·11-s − 3.41·12-s − 0.458·13-s + 2.19·14-s + 0.0252·15-s + 0.856·16-s − 0.364·17-s − 3.52·18-s − 1.15·19-s − 0.0281·20-s + 2.22·21-s − 0.172·22-s − 1.11·23-s + 2.85·24-s − 0.999·25-s + 0.788·26-s − 1.84·27-s − 2.48·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(619\)
Sign: $1$
Analytic conductor: \(4.94273\)
Root analytic conductor: \(2.22322\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 619,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1324394092\)
\(L(\frac12)\) \(\approx\) \(0.1324394092\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad619 \( 1 - T \)
good2 \( 1 + 2.42T + 2T^{2} \)
3 \( 1 + 3.02T + 3T^{2} \)
5 \( 1 + 0.0323T + 5T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 - 0.333T + 11T^{2} \)
13 \( 1 + 1.65T + 13T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + 5.02T + 19T^{2} \)
23 \( 1 + 5.35T + 23T^{2} \)
29 \( 1 - 3.38T + 29T^{2} \)
31 \( 1 + 5.45T + 31T^{2} \)
37 \( 1 + 4.83T + 37T^{2} \)
41 \( 1 - 0.0826T + 41T^{2} \)
43 \( 1 - 1.78T + 43T^{2} \)
47 \( 1 - 7.93T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 9.44T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 - 2.17T + 79T^{2} \)
83 \( 1 - 0.893T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 8.98T + 97T^{2} \)
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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

−10.52637136849796776226669401827, −9.868399526358912553283208111175, −9.198703397501711932983771184150, −7.974964802837478912045468932593, −6.89384940346395954899253171815, −6.47598642968216631844279391624, −5.59302127962994660058418889610, −4.06908770148054944180905901870, −2.10117839316690830052936000301, −0.41928144542520850998814378259, 0.41928144542520850998814378259, 2.10117839316690830052936000301, 4.06908770148054944180905901870, 5.59302127962994660058418889610, 6.47598642968216631844279391624, 6.89384940346395954899253171815, 7.974964802837478912045468932593, 9.198703397501711932983771184150, 9.868399526358912553283208111175, 10.52637136849796776226669401827

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