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Properties

Label 2-2919-2919.1550-c0-0-0
Degree $2$
Conductor $2919$
Sign $0.981 - 0.191i$
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  + (0.803 + 0.595i)3-s + (0.998 − 0.0455i)4-s + (−0.746 − 0.665i)7-s + (0.291 + 0.956i)9-s + (0.829 + 0.557i)12-s + (0.905 − 1.22i)13-s + (0.995 − 0.0909i)16-s + (1.46 + 0.270i)19-s + (−0.203 − 0.979i)21-s + (−0.962 − 0.269i)25-s + (−0.334 + 0.942i)27-s + (−0.775 − 0.631i)28-s + (−1.33 − 0.407i)31-s + (0.334 + 0.942i)36-s + (−1.11 − 0.311i)37-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)3-s + (0.998 − 0.0455i)4-s + (−0.746 − 0.665i)7-s + (0.291 + 0.956i)9-s + (0.829 + 0.557i)12-s + (0.905 − 1.22i)13-s + (0.995 − 0.0909i)16-s + (1.46 + 0.270i)19-s + (−0.203 − 0.979i)21-s + (−0.962 − 0.269i)25-s + (−0.334 + 0.942i)27-s + (−0.775 − 0.631i)28-s + (−1.33 − 0.407i)31-s + (0.334 + 0.942i)36-s + (−1.11 − 0.311i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ 0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.015838720\)
\(L(\frac12)\) \(\approx\) \(2.015838720\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.803 - 0.595i)T \)
7 \( 1 + (0.746 + 0.665i)T \)
139 \( 1 + (0.983 - 0.181i)T \)
good2 \( 1 + (-0.998 + 0.0455i)T^{2} \)
5 \( 1 + (0.962 + 0.269i)T^{2} \)
11 \( 1 + (-0.203 + 0.979i)T^{2} \)
13 \( 1 + (-0.905 + 1.22i)T + (-0.291 - 0.956i)T^{2} \)
17 \( 1 + (0.158 + 0.987i)T^{2} \)
19 \( 1 + (-1.46 - 0.270i)T + (0.934 + 0.356i)T^{2} \)
23 \( 1 + (-0.648 - 0.761i)T^{2} \)
29 \( 1 + (-0.995 + 0.0909i)T^{2} \)
31 \( 1 + (1.33 + 0.407i)T + (0.829 + 0.557i)T^{2} \)
37 \( 1 + (1.11 + 0.311i)T + (0.854 + 0.519i)T^{2} \)
41 \( 1 + (0.803 + 0.595i)T^{2} \)
43 \( 1 + (1.09 - 0.631i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.962 - 0.269i)T^{2} \)
53 \( 1 + (0.538 + 0.842i)T^{2} \)
59 \( 1 + (-0.995 - 0.0909i)T^{2} \)
61 \( 1 + (-1.36 - 1.22i)T + (0.113 + 0.993i)T^{2} \)
67 \( 1 + (0.951 + 0.774i)T + (0.203 + 0.979i)T^{2} \)
71 \( 1 + (0.715 - 0.699i)T^{2} \)
73 \( 1 + (-0.135 - 1.97i)T + (-0.990 + 0.136i)T^{2} \)
79 \( 1 + (-1.66 + 0.549i)T + (0.803 - 0.595i)T^{2} \)
83 \( 1 + (-0.949 + 0.313i)T^{2} \)
89 \( 1 + (0.854 - 0.519i)T^{2} \)
97 \( 1 + (0.998 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.019156495217057477353150644865, −8.025815509096689486731613867005, −7.60045849928869649168212102357, −6.84957379538100256893043687574, −5.85455927776867714779348150243, −5.24655963139812756951439788647, −3.72534993674942357535218198858, −3.50028268571058978123666583164, −2.58117598614042332035370893898, −1.36237471244455430830828191816, 1.53698634814569050864735019180, 2.19369784989169332385357869480, 3.34351734240472476102408339364, 3.63067340922796980623182208010, 5.29191825998274476553460684164, 6.12551211188411643590068311385, 6.79042587138019240462091014813, 7.26075977469568133507093804240, 8.137490816941344708670041130373, 8.945436581049058855552987368678

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