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Properties

Label 2-2940-105.104-c1-0-25
Degree $2$
Conductor $2940$
Sign $0.758 + 0.651i$
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  + (−0.299 − 1.70i)3-s + (−1.67 − 1.48i)5-s + (−2.82 + 1.02i)9-s + 6.04i·11-s − 2.23·13-s + (−2.03 + 3.29i)15-s − 2.64i·17-s − 3.11i·19-s + 3.68·23-s + (0.592 + 4.96i)25-s + (2.58 + 4.50i)27-s + 5.25i·29-s + 4.70i·31-s + (10.3 − 1.80i)33-s − 6.75i·37-s + ⋯
L(s)  = 1  + (−0.172 − 0.984i)3-s + (−0.747 − 0.663i)5-s + (−0.940 + 0.340i)9-s + 1.82i·11-s − 0.620·13-s + (−0.524 + 0.851i)15-s − 0.640i·17-s − 0.713i·19-s + 0.769·23-s + (0.118 + 0.992i)25-s + (0.497 + 0.867i)27-s + 0.976i·29-s + 0.844i·31-s + (1.79 − 0.314i)33-s − 1.11i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.758 + 0.651i$
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: \((2,\ 2940,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.154058953\)
\(L(\frac12)\) \(\approx\) \(1.154058953\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.299 + 1.70i)T \)
5 \( 1 + (1.67 + 1.48i)T \)
7 \( 1 \)
good11 \( 1 - 6.04iT - 11T^{2} \)
13 \( 1 + 2.23T + 13T^{2} \)
17 \( 1 + 2.64iT - 17T^{2} \)
19 \( 1 + 3.11iT - 19T^{2} \)
23 \( 1 - 3.68T + 23T^{2} \)
29 \( 1 - 5.25iT - 29T^{2} \)
31 \( 1 - 4.70iT - 31T^{2} \)
37 \( 1 + 6.75iT - 37T^{2} \)
41 \( 1 - 0.179T + 41T^{2} \)
43 \( 1 + 11.7iT - 43T^{2} \)
47 \( 1 - 7.14iT - 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 - 0.839iT - 71T^{2} \)
73 \( 1 + 3.80T + 73T^{2} \)
79 \( 1 + 0.107T + 79T^{2} \)
83 \( 1 - 8.52iT - 83T^{2} \)
89 \( 1 - 5.52T + 89T^{2} \)
97 \( 1 - 8.41T + 97T^{2} \)
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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

−8.720742704854805837815193425596, −7.64094009365961220711137655529, −7.13949052361922206333463927239, −6.86183500151107866681150624104, −5.29729731414838363226439015500, −5.02458874838447084636539646483, −4.02824901312327212388776254502, −2.76577919893507873871043615602, −1.87330065620029638246530530134, −0.69980186045769938265778366716, 0.60716998933602878763544488995, 2.59848190743511317277100152746, 3.37188651147554402659020565948, 3.96060382671571209474156516374, 4.86989822886677494903446720512, 5.89759141286376297181729408703, 6.29376809284573661689082980211, 7.45005749226013050046451918155, 8.282966016344078918503005633043, 8.664384503527730249846995841118

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