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Properties

Label 2-2919-2919.1361-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.225 - 0.974i$
Analytic cond. $1.45677$
Root an. cond. $1.20696$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.291 − 0.956i)3-s + (−0.995 − 0.0909i)4-s + (−0.113 + 0.993i)7-s + (−0.829 − 0.557i)9-s + (−0.377 + 0.926i)12-s + (−1.88 − 0.575i)13-s + (0.983 + 0.181i)16-s + (0.212 − 0.0809i)19-s + (0.917 + 0.398i)21-s + (−0.854 + 0.519i)25-s + (−0.775 + 0.631i)27-s + (0.203 − 0.979i)28-s + (1.11 + 1.65i)31-s + (0.775 + 0.631i)36-s + (−0.572 + 0.347i)37-s + ⋯
L(s)  = 1  + (0.291 − 0.956i)3-s + (−0.995 − 0.0909i)4-s + (−0.113 + 0.993i)7-s + (−0.829 − 0.557i)9-s + (−0.377 + 0.926i)12-s + (−1.88 − 0.575i)13-s + (0.983 + 0.181i)16-s + (0.212 − 0.0809i)19-s + (0.917 + 0.398i)21-s + (−0.854 + 0.519i)25-s + (−0.775 + 0.631i)27-s + (0.203 − 0.979i)28-s + (1.11 + 1.65i)31-s + (0.775 + 0.631i)36-s + (−0.572 + 0.347i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.225 - 0.974i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.225 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2899058067\)
\(L(\frac12)\) \(\approx\) \(0.2899058067\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.291 + 0.956i)T \)
7 \( 1 + (0.113 - 0.993i)T \)
139 \( 1 + (0.934 + 0.356i)T \)
good2 \( 1 + (0.995 + 0.0909i)T^{2} \)
5 \( 1 + (0.854 - 0.519i)T^{2} \)
11 \( 1 + (0.917 - 0.398i)T^{2} \)
13 \( 1 + (1.88 + 0.575i)T + (0.829 + 0.557i)T^{2} \)
17 \( 1 + (0.949 + 0.313i)T^{2} \)
19 \( 1 + (-0.212 + 0.0809i)T + (0.746 - 0.665i)T^{2} \)
23 \( 1 + (-0.158 - 0.987i)T^{2} \)
29 \( 1 + (-0.983 - 0.181i)T^{2} \)
31 \( 1 + (-1.11 - 1.65i)T + (-0.377 + 0.926i)T^{2} \)
37 \( 1 + (0.572 - 0.347i)T + (0.460 - 0.887i)T^{2} \)
41 \( 1 + (0.291 - 0.956i)T^{2} \)
43 \( 1 + (1.69 - 0.979i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.854 + 0.519i)T^{2} \)
53 \( 1 + (-0.419 - 0.907i)T^{2} \)
59 \( 1 + (-0.983 + 0.181i)T^{2} \)
61 \( 1 + (0.155 - 1.35i)T + (-0.974 - 0.225i)T^{2} \)
67 \( 1 + (0.100 - 0.485i)T + (-0.917 - 0.398i)T^{2} \)
71 \( 1 + (-0.0227 - 0.999i)T^{2} \)
73 \( 1 + (1.90 + 0.262i)T + (0.962 + 0.269i)T^{2} \)
79 \( 1 + (-0.866 - 0.641i)T + (0.291 + 0.956i)T^{2} \)
83 \( 1 + (0.803 + 0.595i)T^{2} \)
89 \( 1 + (0.460 + 0.887i)T^{2} \)
97 \( 1 + (-0.995 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.984198774521433027912193533366, −8.440193023956566998040276981514, −7.75129871530621314384977915750, −7.01683044918293351942295510036, −6.06197736853511209834335150798, −5.27782259599552328762194741971, −4.74437336027408780018872363966, −3.30004043052447872284204475205, −2.67781124247932501389482845563, −1.49601787256480177401653949145, 0.17503609709930486624610374675, 2.18885732139112036789164714613, 3.34245445548722085748119515835, 4.12025590811185708957003938851, 4.67316083534575936370785919642, 5.28528468689227071578744850609, 6.41748757832451986606298946803, 7.53962929311817805425591699499, 7.951584935275634528860053579850, 8.893160807907243018376800373210

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