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Properties

Label 2-2919-2919.698-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.673 + 0.739i$
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  + (−0.998 − 0.0455i)3-s + (0.974 − 0.225i)4-s + (0.877 + 0.480i)7-s + (0.995 + 0.0909i)9-s + (−0.983 + 0.181i)12-s + (0.0842 − 1.85i)13-s + (0.898 − 0.439i)16-s + (−1.07 − 1.38i)19-s + (−0.854 − 0.519i)21-s + (−0.203 − 0.979i)25-s + (−0.990 − 0.136i)27-s + (0.962 + 0.269i)28-s + (0.121 + 1.32i)31-s + (0.990 − 0.136i)36-s + (−0.0277 − 0.133i)37-s + ⋯
L(s)  = 1  + (−0.998 − 0.0455i)3-s + (0.974 − 0.225i)4-s + (0.877 + 0.480i)7-s + (0.995 + 0.0909i)9-s + (−0.983 + 0.181i)12-s + (0.0842 − 1.85i)13-s + (0.898 − 0.439i)16-s + (−1.07 − 1.38i)19-s + (−0.854 − 0.519i)21-s + (−0.203 − 0.979i)25-s + (−0.990 − 0.136i)27-s + (0.962 + 0.269i)28-s + (0.121 + 1.32i)31-s + (0.990 − 0.136i)36-s + (−0.0277 − 0.133i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239651563\)
\(L(\frac12)\) \(\approx\) \(1.239651563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.998 + 0.0455i)T \)
7 \( 1 + (-0.877 - 0.480i)T \)
139 \( 1 + (0.613 - 0.789i)T \)
good2 \( 1 + (-0.974 + 0.225i)T^{2} \)
5 \( 1 + (0.203 + 0.979i)T^{2} \)
11 \( 1 + (-0.854 + 0.519i)T^{2} \)
13 \( 1 + (-0.0842 + 1.85i)T + (-0.995 - 0.0909i)T^{2} \)
17 \( 1 + (0.715 + 0.699i)T^{2} \)
19 \( 1 + (1.07 + 1.38i)T + (-0.247 + 0.968i)T^{2} \)
23 \( 1 + (0.377 + 0.926i)T^{2} \)
29 \( 1 + (-0.898 + 0.439i)T^{2} \)
31 \( 1 + (-0.121 - 1.32i)T + (-0.983 + 0.181i)T^{2} \)
37 \( 1 + (0.0277 + 0.133i)T + (-0.917 + 0.398i)T^{2} \)
41 \( 1 + (-0.998 - 0.0455i)T^{2} \)
43 \( 1 + (0.467 + 0.269i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.203 - 0.979i)T^{2} \)
53 \( 1 + (0.291 - 0.956i)T^{2} \)
59 \( 1 + (-0.898 - 0.439i)T^{2} \)
61 \( 1 + (-0.807 - 0.441i)T + (0.538 + 0.842i)T^{2} \)
67 \( 1 + (0.305 + 0.0856i)T + (0.854 + 0.519i)T^{2} \)
71 \( 1 + (-0.746 + 0.665i)T^{2} \)
73 \( 1 + (-0.519 - 1.46i)T + (-0.775 + 0.631i)T^{2} \)
79 \( 1 + (-0.0365 - 1.60i)T + (-0.998 + 0.0455i)T^{2} \)
83 \( 1 + (0.0227 + 0.999i)T^{2} \)
89 \( 1 + (-0.917 - 0.398i)T^{2} \)
97 \( 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.609638616832615858786544326192, −8.061769616956006670750414785223, −7.14454758338943525927101672407, −6.56759661003327891073892565798, −5.66913859191313602224913238042, −5.26060552243134243833847108390, −4.36921594784759851599227885413, −2.97856788886183353530357353102, −2.11117252517664338578725898331, −0.917437332102453794518304980633, 1.54199037107712359051442382533, 2.02626115070642108239181077533, 3.76601454165191168103179091203, 4.27502936036807595791965138552, 5.25495750618040388491454451274, 6.22249738191600143569541046668, 6.61101423961808756207653714763, 7.48565220522275323388461652961, 7.995072328665809587426845481478, 9.081426658710268014727240761376

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