Properties

Label 2-619-1.1-c1-0-49
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.32·2-s + 0.433·3-s − 0.244·4-s − 1.11·5-s + 0.574·6-s − 1.01·7-s − 2.97·8-s − 2.81·9-s − 1.47·10-s − 3.85·11-s − 0.106·12-s − 5.34·13-s − 1.34·14-s − 0.482·15-s − 3.45·16-s + 6.52·17-s − 3.72·18-s + 0.174·19-s + 0.272·20-s − 0.438·21-s − 5.11·22-s + 4.72·23-s − 1.28·24-s − 3.75·25-s − 7.07·26-s − 2.51·27-s + 0.247·28-s + ⋯
L(s)  = 1  + 0.936·2-s + 0.250·3-s − 0.122·4-s − 0.498·5-s + 0.234·6-s − 0.382·7-s − 1.05·8-s − 0.937·9-s − 0.466·10-s − 1.16·11-s − 0.0306·12-s − 1.48·13-s − 0.358·14-s − 0.124·15-s − 0.862·16-s + 1.58·17-s − 0.878·18-s + 0.0399·19-s + 0.0610·20-s − 0.0956·21-s − 1.09·22-s + 0.984·23-s − 0.263·24-s − 0.751·25-s − 1.38·26-s − 0.484·27-s + 0.0468·28-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: \(1\)
Selberg data: \((2,\ 619,\ (\ :1/2),\ -1)\)

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)$
bad619 \( 1 + T \)
good2 \( 1 - 1.32T + 2T^{2} \)
3 \( 1 - 0.433T + 3T^{2} \)
5 \( 1 + 1.11T + 5T^{2} \)
7 \( 1 + 1.01T + 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + 5.34T + 13T^{2} \)
17 \( 1 - 6.52T + 17T^{2} \)
19 \( 1 - 0.174T + 19T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 - 1.79T + 31T^{2} \)
37 \( 1 - 3.31T + 37T^{2} \)
41 \( 1 + 1.64T + 41T^{2} \)
43 \( 1 + 0.534T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + 1.75T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 + 9.54T + 71T^{2} \)
73 \( 1 - 5.76T + 73T^{2} \)
79 \( 1 - 8.04T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.06414293244683573791800608124, −9.446343429852388846618501396065, −8.222653050907202998543997859681, −7.67387295214651135385956355518, −6.33359776496913207900171225340, −5.29711363508147175128727311369, −4.74198248094815234656620710356, −3.26056674085726468459806085525, −2.81109235008017523712803848187, 0, 2.81109235008017523712803848187, 3.26056674085726468459806085525, 4.74198248094815234656620710356, 5.29711363508147175128727311369, 6.33359776496913207900171225340, 7.67387295214651135385956355518, 8.222653050907202998543997859681, 9.446343429852388846618501396065, 10.06414293244683573791800608124

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