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Properties

Label 2-2919-2919.509-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.801 - 0.598i$
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  + (−0.158 − 0.987i)3-s + (0.715 − 0.699i)4-s + (−0.983 + 0.181i)7-s + (−0.949 + 0.313i)9-s + (−0.803 − 0.595i)12-s + (−1.66 + 0.267i)13-s + (0.0227 − 0.999i)16-s + (−1.96 + 0.0895i)19-s + (0.334 + 0.942i)21-s + (0.0682 + 0.997i)25-s + (0.460 + 0.887i)27-s + (−0.576 + 0.816i)28-s + (−0.349 + 1.05i)31-s + (−0.460 + 0.887i)36-s + (−0.116 − 1.70i)37-s + ⋯
L(s)  = 1  + (−0.158 − 0.987i)3-s + (0.715 − 0.699i)4-s + (−0.983 + 0.181i)7-s + (−0.949 + 0.313i)9-s + (−0.803 − 0.595i)12-s + (−1.66 + 0.267i)13-s + (0.0227 − 0.999i)16-s + (−1.96 + 0.0895i)19-s + (0.334 + 0.942i)21-s + (0.0682 + 0.997i)25-s + (0.460 + 0.887i)27-s + (−0.576 + 0.816i)28-s + (−0.349 + 1.05i)31-s + (−0.460 + 0.887i)36-s + (−0.116 − 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.801 - 0.598i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2945544869\)
\(L(\frac12)\) \(\approx\) \(0.2945544869\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.158 + 0.987i)T \)
7 \( 1 + (0.983 - 0.181i)T \)
139 \( 1 + (-0.998 - 0.0455i)T \)
good2 \( 1 + (-0.715 + 0.699i)T^{2} \)
5 \( 1 + (-0.0682 - 0.997i)T^{2} \)
11 \( 1 + (0.334 - 0.942i)T^{2} \)
13 \( 1 + (1.66 - 0.267i)T + (0.949 - 0.313i)T^{2} \)
17 \( 1 + (0.419 - 0.907i)T^{2} \)
19 \( 1 + (1.96 - 0.0895i)T + (0.995 - 0.0909i)T^{2} \)
23 \( 1 + (0.538 - 0.842i)T^{2} \)
29 \( 1 + (-0.0227 + 0.999i)T^{2} \)
31 \( 1 + (0.349 - 1.05i)T + (-0.803 - 0.595i)T^{2} \)
37 \( 1 + (0.116 + 1.70i)T + (-0.990 + 0.136i)T^{2} \)
41 \( 1 + (-0.158 - 0.987i)T^{2} \)
43 \( 1 + (1.41 + 0.816i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.0682 + 0.997i)T^{2} \)
53 \( 1 + (-0.247 - 0.968i)T^{2} \)
59 \( 1 + (-0.0227 - 0.999i)T^{2} \)
61 \( 1 + (-1.52 + 0.280i)T + (0.934 - 0.356i)T^{2} \)
67 \( 1 + (-1.12 + 1.59i)T + (-0.334 - 0.942i)T^{2} \)
71 \( 1 + (0.829 - 0.557i)T^{2} \)
73 \( 1 + (1.25 + 0.543i)T + (0.682 + 0.730i)T^{2} \)
79 \( 1 + (0.795 + 0.933i)T + (-0.158 + 0.987i)T^{2} \)
83 \( 1 + (-0.648 - 0.761i)T^{2} \)
89 \( 1 + (-0.990 - 0.136i)T^{2} \)
97 \( 1 + (0.715 + 1.23i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.494290067457268779187612476408, −7.30048569905987081418134875845, −7.02889439229064448947605948089, −6.34803073777512694851160574151, −5.61517572235256070992081092306, −4.86757549611654686792578384855, −3.42250619407315290433415357014, −2.37667763998781340971459802372, −1.87672619529256218337184799514, −0.15938128577435758715649329135, 2.39236191325603449230953793666, 2.87630508009035663781788514647, 3.94736223209199897657746834032, 4.50920509093185879363445334112, 5.59190556722073300240197834309, 6.54126095293206844157231552331, 6.88823079906732432205760295412, 8.079863315555851780036775130495, 8.532844421595710135540302532399, 9.612764242874749088959079676637

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