Properties

Label 2-619-1.1-c1-0-27
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  − 1.37·2-s + 0.756·3-s − 0.118·4-s − 1.92·5-s − 1.03·6-s − 0.911·7-s + 2.90·8-s − 2.42·9-s + 2.64·10-s + 6.12·11-s − 0.0895·12-s + 0.967·13-s + 1.25·14-s − 1.45·15-s − 3.74·16-s + 1.00·17-s + 3.32·18-s − 1.01·19-s + 0.228·20-s − 0.689·21-s − 8.40·22-s − 8.98·23-s + 2.19·24-s − 1.28·25-s − 1.32·26-s − 4.10·27-s + 0.107·28-s + ⋯
L(s)  = 1  − 0.969·2-s + 0.436·3-s − 0.0591·4-s − 0.861·5-s − 0.423·6-s − 0.344·7-s + 1.02·8-s − 0.809·9-s + 0.835·10-s + 1.84·11-s − 0.0258·12-s + 0.268·13-s + 0.334·14-s − 0.376·15-s − 0.937·16-s + 0.243·17-s + 0.784·18-s − 0.233·19-s + 0.0509·20-s − 0.150·21-s − 1.79·22-s − 1.87·23-s + 0.448·24-s − 0.257·25-s − 0.260·26-s − 0.790·27-s + 0.0203·28-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 \( 1 + 1.37T + 2T^{2} \)
3 \( 1 - 0.756T + 3T^{2} \)
5 \( 1 + 1.92T + 5T^{2} \)
7 \( 1 + 0.911T + 7T^{2} \)
11 \( 1 - 6.12T + 11T^{2} \)
13 \( 1 - 0.967T + 13T^{2} \)
17 \( 1 - 1.00T + 17T^{2} \)
19 \( 1 + 1.01T + 19T^{2} \)
23 \( 1 + 8.98T + 23T^{2} \)
29 \( 1 + 8.45T + 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 - 7.02T + 37T^{2} \)
41 \( 1 + 1.86T + 41T^{2} \)
43 \( 1 + 0.388T + 43T^{2} \)
47 \( 1 + 7.67T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + 0.806T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 7.03T + 67T^{2} \)
71 \( 1 + 6.02T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 1.04T + 83T^{2} \)
89 \( 1 + 4.36T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−9.677824742406174367188481423895, −9.423108029950084811500781911324, −8.357484750600153294910299487694, −7.974608061985666124757097785957, −6.86995230096114187637285444899, −5.79988351811717877234950239474, −4.12991269655606048862329023328, −3.62183402417386229032307429190, −1.75183275566667485948469303355, 0, 1.75183275566667485948469303355, 3.62183402417386229032307429190, 4.12991269655606048862329023328, 5.79988351811717877234950239474, 6.86995230096114187637285444899, 7.974608061985666124757097785957, 8.357484750600153294910299487694, 9.423108029950084811500781911324, 9.677824742406174367188481423895

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