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Properties

Label 2-2940-35.27-c1-0-3
Degree $2$
Conductor $2940$
Sign $-0.0462 - 0.998i$
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  + (−0.707 − 0.707i)3-s + (1.71 − 1.43i)5-s + 1.00i·9-s − 3.14·11-s + (−4.55 − 4.55i)13-s + (−2.22 − 0.191i)15-s + (−2.07 + 2.07i)17-s − 2.98·19-s + (1.01 − 1.01i)23-s + (0.853 − 4.92i)25-s + (0.707 − 0.707i)27-s + 9.49i·29-s + 2.89i·31-s + (2.22 + 2.22i)33-s + (7.64 + 7.64i)37-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.765 − 0.643i)5-s + 0.333i·9-s − 0.948·11-s + (−1.26 − 1.26i)13-s + (−0.575 − 0.0494i)15-s + (−0.503 + 0.503i)17-s − 0.684·19-s + (0.211 − 0.211i)23-s + (0.170 − 0.985i)25-s + (0.136 − 0.136i)27-s + 1.76i·29-s + 0.519i·31-s + (0.387 + 0.387i)33-s + (1.25 + 1.25i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0462 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0462 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.0462 - 0.998i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.0462 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4791785676\)
\(L(\frac12)\) \(\approx\) \(0.4791785676\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.71 + 1.43i)T \)
7 \( 1 \)
good11 \( 1 + 3.14T + 11T^{2} \)
13 \( 1 + (4.55 + 4.55i)T + 13iT^{2} \)
17 \( 1 + (2.07 - 2.07i)T - 17iT^{2} \)
19 \( 1 + 2.98T + 19T^{2} \)
23 \( 1 + (-1.01 + 1.01i)T - 23iT^{2} \)
29 \( 1 - 9.49iT - 29T^{2} \)
31 \( 1 - 2.89iT - 31T^{2} \)
37 \( 1 + (-7.64 - 7.64i)T + 37iT^{2} \)
41 \( 1 - 5.93iT - 41T^{2} \)
43 \( 1 + (7.74 - 7.74i)T - 43iT^{2} \)
47 \( 1 + (-3.00 + 3.00i)T - 47iT^{2} \)
53 \( 1 + (-0.567 + 0.567i)T - 53iT^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 1.49iT - 61T^{2} \)
67 \( 1 + (2.56 + 2.56i)T + 67iT^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + (-4.09 - 4.09i)T + 73iT^{2} \)
79 \( 1 - 2.90iT - 79T^{2} \)
83 \( 1 + (7.47 + 7.47i)T + 83iT^{2} \)
89 \( 1 - 7.47T + 89T^{2} \)
97 \( 1 + (-7.36 + 7.36i)T - 97iT^{2} \)
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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

−8.764066494072015161213012264293, −8.242619266428113700366064360345, −7.46378317876758989516555630203, −6.58880868182407533692724448508, −5.85941504381371509376127084349, −4.99965194839974364371791152200, −4.73535429000080887148364135690, −3.06638145845913008717988356245, −2.30691629925839996691848045725, −1.17026831499456868957226869215, 0.15880261220821081535094299106, 2.17603544146853234724922146511, 2.51923130493402038789065877279, 3.95007947282614778855708751789, 4.69414357098475344558378003183, 5.52391792680944570197569636159, 6.20636538421150858295834363381, 7.05973199823799514772627015331, 7.55802494885095034849223202854, 8.771792570306996975977730496077

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