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Properties

Label 2-2997-333.248-c0-0-1
Degree $2$
Conductor $2997$
Sign $0.476 - 0.879i$
Analytic cond. $1.49569$
Root an. cond. $1.22298$
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  + (−0.5 − 0.866i)4-s − 7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + 37-s + (0.5 + 0.866i)43-s + (0.499 − 0.866i)52-s + 2·61-s + 0.999·64-s + (0.5 + 0.866i)67-s − 73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s − 7-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s + (−1 + 1.73i)19-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)28-s + (0.5 − 0.866i)31-s + 37-s + (0.5 + 0.866i)43-s + (0.499 − 0.866i)52-s + 2·61-s + 0.999·64-s + (0.5 + 0.866i)67-s − 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2997 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2997\)    =    \(3^{4} \cdot 37\)
Sign: $0.476 - 0.879i$
Analytic conductor: \(1.49569\)
Root analytic conductor: \(1.22298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2997} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2997,\ (\ :0),\ 0.476 - 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7044813987\)
\(L(\frac12)\) \(\approx\) \(0.7044813987\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 - T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.143989912223505842888152812567, −8.450372822113304221573681069535, −7.56743071644568116441682550012, −6.34236473950409669113035971498, −6.24184530062331045619715218241, −5.33319821965066345893196067287, −4.17525700175545161777242501269, −3.77109961171108674962897957087, −2.36431669013324420721161286795, −1.28944636173333424308431038453, 0.47119963947046325619836032752, 2.49016188876196963373007391222, 3.12670817748843435937452923737, 4.02517247548779917919456924635, 4.74921343761798913427077214264, 5.79201103829764136856937153873, 6.64273443456031343122031758493, 7.22634281189880283515893826136, 8.266750569194536223805670723348, 8.636407904762389856567143838343

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