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Properties

Label 2-2919-2919.521-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.779 + 0.626i$
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  + (−0.974 − 0.225i)3-s + (0.419 − 0.907i)4-s + (−0.803 + 0.595i)7-s + (0.898 + 0.439i)9-s + (−0.613 + 0.789i)12-s + (−0.160 + 0.694i)13-s + (−0.648 − 0.761i)16-s + (−0.255 − 1.58i)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 + (−0.422 − 0.862i)31-s + (0.775 − 0.631i)36-s + (−0.572 − 0.347i)37-s + ⋯
L(s)  = 1  + (−0.974 − 0.225i)3-s + (0.419 − 0.907i)4-s + (−0.803 + 0.595i)7-s + (0.898 + 0.439i)9-s + (−0.613 + 0.789i)12-s + (−0.160 + 0.694i)13-s + (−0.648 − 0.761i)16-s + (−0.255 − 1.58i)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 + (−0.422 − 0.862i)31-s + (0.775 − 0.631i)36-s + (−0.572 − 0.347i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.779 + 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5259225888\)
\(L(\frac12)\) \(\approx\) \(0.5259225888\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 + 0.225i)T \)
7 \( 1 + (0.803 - 0.595i)T \)
139 \( 1 + (-0.158 + 0.987i)T \)
good2 \( 1 + (-0.419 + 0.907i)T^{2} \)
5 \( 1 + (0.854 + 0.519i)T^{2} \)
11 \( 1 + (0.917 + 0.398i)T^{2} \)
13 \( 1 + (0.160 - 0.694i)T + (-0.898 - 0.439i)T^{2} \)
17 \( 1 + (-0.746 - 0.665i)T^{2} \)
19 \( 1 + (0.255 + 1.58i)T + (-0.949 + 0.313i)T^{2} \)
23 \( 1 + (0.934 - 0.356i)T^{2} \)
29 \( 1 + (0.648 + 0.761i)T^{2} \)
31 \( 1 + (0.422 + 0.862i)T + (-0.613 + 0.789i)T^{2} \)
37 \( 1 + (0.572 + 0.347i)T + (0.460 + 0.887i)T^{2} \)
41 \( 1 + (-0.974 - 0.225i)T^{2} \)
43 \( 1 + (-1.69 + 0.979i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.854 - 0.519i)T^{2} \)
53 \( 1 + (0.995 - 0.0909i)T^{2} \)
59 \( 1 + (0.648 - 0.761i)T^{2} \)
61 \( 1 + (1.09 - 0.812i)T + (0.291 - 0.956i)T^{2} \)
67 \( 1 + (0.290 + 1.40i)T + (-0.917 + 0.398i)T^{2} \)
71 \( 1 + (0.877 - 0.480i)T^{2} \)
73 \( 1 + (1.90 - 0.262i)T + (0.962 - 0.269i)T^{2} \)
79 \( 1 + (0.226 + 1.98i)T + (-0.974 + 0.225i)T^{2} \)
83 \( 1 + (0.113 + 0.993i)T^{2} \)
89 \( 1 + (0.460 - 0.887i)T^{2} \)
97 \( 1 + (0.419 - 0.726i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.035481528170111208094604475080, −7.52483706312873457725282164218, −6.97504259981106128300403229456, −6.20584711208394075194030653582, −5.81573745116302304641285661456, −4.94155907472000559348088801557, −4.13862471242596687196042743549, −2.63344921214302952163074649633, −1.85686675235168769595366437002, −0.36139965831510071289748056552, 1.50263122766963469869179222827, 2.98388994181604462922680425702, 3.75028397004576964643315450392, 4.38405712136043514719335819973, 5.63271791308787198775264099121, 6.14085593785340295427830132275, 7.04355992702700799341292208058, 7.53040091545278178484846707690, 8.333896871887147595709343232441, 9.395996452866478619272141703626

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