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Properties

Label 2-2919-2919.1769-c0-0-0
Degree $2$
Conductor $2919$
Sign $-0.215 + 0.976i$
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  + (−0.715 + 0.699i)3-s + (−0.746 − 0.665i)4-s + (−0.613 − 0.789i)7-s + (0.0227 − 0.999i)9-s + (0.998 − 0.0455i)12-s + (1.33 + 1.36i)13-s + (0.113 + 0.993i)16-s + (−1.19 − 0.276i)19-s + (0.990 + 0.136i)21-s + (0.334 − 0.942i)25-s + (0.682 + 0.730i)27-s + (−0.0682 + 0.997i)28-s + (−0.362 + 0.00824i)31-s + (−0.682 + 0.730i)36-s + (0.614 − 1.72i)37-s + ⋯
L(s)  = 1  + (−0.715 + 0.699i)3-s + (−0.746 − 0.665i)4-s + (−0.613 − 0.789i)7-s + (0.0227 − 0.999i)9-s + (0.998 − 0.0455i)12-s + (1.33 + 1.36i)13-s + (0.113 + 0.993i)16-s + (−1.19 − 0.276i)19-s + (0.990 + 0.136i)21-s + (0.334 − 0.942i)25-s + (0.682 + 0.730i)27-s + (−0.0682 + 0.997i)28-s + (−0.362 + 0.00824i)31-s + (−0.682 + 0.730i)36-s + (0.614 − 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2919 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2919\)    =    \(3 \cdot 7 \cdot 139\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(1.45677\)
Root analytic conductor: \(1.20696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2919} (1769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2919,\ (\ :0),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4623287391\)
\(L(\frac12)\) \(\approx\) \(0.4623287391\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.715 - 0.699i)T \)
7 \( 1 + (0.613 + 0.789i)T \)
139 \( 1 + (-0.974 + 0.225i)T \)
good2 \( 1 + (0.746 + 0.665i)T^{2} \)
5 \( 1 + (-0.334 + 0.942i)T^{2} \)
11 \( 1 + (0.990 - 0.136i)T^{2} \)
13 \( 1 + (-1.33 - 1.36i)T + (-0.0227 + 0.999i)T^{2} \)
17 \( 1 + (0.829 - 0.557i)T^{2} \)
19 \( 1 + (1.19 + 0.276i)T + (0.898 + 0.439i)T^{2} \)
23 \( 1 + (0.291 - 0.956i)T^{2} \)
29 \( 1 + (-0.113 - 0.993i)T^{2} \)
31 \( 1 + (0.362 - 0.00824i)T + (0.998 - 0.0455i)T^{2} \)
37 \( 1 + (-0.614 + 1.72i)T + (-0.775 - 0.631i)T^{2} \)
41 \( 1 + (-0.715 + 0.699i)T^{2} \)
43 \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.334 - 0.942i)T^{2} \)
53 \( 1 + (-0.949 + 0.313i)T^{2} \)
59 \( 1 + (-0.113 + 0.993i)T^{2} \)
61 \( 1 + (1.18 + 1.52i)T + (-0.247 + 0.968i)T^{2} \)
67 \( 1 + (-0.0572 + 0.836i)T + (-0.990 - 0.136i)T^{2} \)
71 \( 1 + (-0.983 + 0.181i)T^{2} \)
73 \( 1 + (0.530 + 1.02i)T + (-0.576 + 0.816i)T^{2} \)
79 \( 1 + (0.119 - 0.293i)T + (-0.715 - 0.699i)T^{2} \)
83 \( 1 + (0.377 - 0.926i)T^{2} \)
89 \( 1 + (-0.775 + 0.631i)T^{2} \)
97 \( 1 + (-0.746 - 1.29i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.106061094928925913909268692940, −8.207784863204562077356179620417, −6.76299672321161472495894825955, −6.44675650422207691507663870132, −5.70820886287979591179487915080, −4.64015770953916276130791255491, −4.15835142184046749171877583072, −3.52537762302426040687421604417, −1.72046874658083857328185198299, −0.35979303647183079481178521453, 1.28484338478429865814496800159, 2.76263714574199536518268427822, 3.46740676588678815915112757152, 4.59999733229109969202080256289, 5.46825323005874400911750288271, 6.06941312250328563182922440806, 6.78334017163762370999167010177, 7.81561560041492088937821274223, 8.374635165756905270559149052394, 8.867913955618618974776126951775

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