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Properties

Label 2-2940-105.104-c1-0-23
Degree $2$
Conductor $2940$
Sign $0.218 - 0.975i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 + 0.367i)3-s + (−2.03 + 0.916i)5-s + (2.73 − 1.24i)9-s + 5.87i·11-s + 3.54·13-s + (3.11 − 2.30i)15-s + 3.10i·17-s − 4.13i·19-s + 7.15·23-s + (3.31 − 3.74i)25-s + (−4.16 + 3.10i)27-s − 6.84i·29-s − 4.50i·31-s + (−2.15 − 9.94i)33-s − 1.97i·37-s + ⋯
L(s)  = 1  + (−0.977 + 0.212i)3-s + (−0.912 + 0.410i)5-s + (0.910 − 0.414i)9-s + 1.77i·11-s + 0.981·13-s + (0.804 − 0.594i)15-s + 0.753i·17-s − 0.949i·19-s + 1.49·23-s + (0.663 − 0.748i)25-s + (−0.801 + 0.598i)27-s − 1.27i·29-s − 0.809i·31-s + (−0.375 − 1.73i)33-s − 0.324i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.060586199\)
\(L(\frac12)\) \(\approx\) \(1.060586199\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.69 - 0.367i)T \)
5 \( 1 + (2.03 - 0.916i)T \)
7 \( 1 \)
good11 \( 1 - 5.87iT - 11T^{2} \)
13 \( 1 - 3.54T + 13T^{2} \)
17 \( 1 - 3.10iT - 17T^{2} \)
19 \( 1 + 4.13iT - 19T^{2} \)
23 \( 1 - 7.15T + 23T^{2} \)
29 \( 1 + 6.84iT - 29T^{2} \)
31 \( 1 + 4.50iT - 31T^{2} \)
37 \( 1 + 1.97iT - 37T^{2} \)
41 \( 1 + 6.33T + 41T^{2} \)
43 \( 1 - 3.88iT - 43T^{2} \)
47 \( 1 - 1.83iT - 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 4.65T + 59T^{2} \)
61 \( 1 + 0.810iT - 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + 1.18iT - 71T^{2} \)
73 \( 1 - 2.17T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 - 5.31iT - 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 - 4.50T + 97T^{2} \)
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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

−8.989036182886567103542592438048, −8.010297396025669921276802877902, −7.18890488066877835109941129394, −6.77464955221581184543084828411, −5.93231954869203454795330727573, −4.86249181010869725925315083283, −4.33347459605591391594916905550, −3.57422465346706983757749447985, −2.24780392304568515856424397288, −0.886590088444976216693112056367, 0.58110988641513348539187522605, 1.34665617538885210065914657804, 3.22059500036454209796720070262, 3.72811026762094370843393870125, 4.95665387742961683230101815470, 5.40227355187155571022755707229, 6.32308799794558887204895679826, 7.00103523096291061779692207610, 7.82051555757784668852757815309, 8.654939717594823821517001756031

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