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Properties

Label 2-2940-35.9-c1-0-1
Degree $2$
Conductor $2940$
Sign $-0.657 - 0.753i$
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  + (−0.866 + 0.5i)3-s + (0.133 − 2.23i)5-s + (0.499 − 0.866i)9-s + (−2 − 3.46i)11-s + 2i·13-s + (1 + 1.99i)15-s + (−1.73 + i)17-s + (1 − 1.73i)19-s + (−5.19 − 3i)23-s + (−4.96 − 0.598i)25-s + 0.999i·27-s − 6·29-s + (3 + 5.19i)31-s + (3.46 + 1.99i)33-s + (3.46 + 2i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.0599 − 0.998i)5-s + (0.166 − 0.288i)9-s + (−0.603 − 1.04i)11-s + 0.554i·13-s + (0.258 + 0.516i)15-s + (−0.420 + 0.242i)17-s + (0.229 − 0.397i)19-s + (−1.08 − 0.625i)23-s + (−0.992 − 0.119i)25-s + 0.192i·27-s − 1.11·29-s + (0.538 + 0.933i)31-s + (0.603 + 0.348i)33-s + (0.569 + 0.328i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1696506218\)
\(L(\frac12)\) \(\approx\) \(0.1696506218\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.133 + 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (-3.46 - 2i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 + i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-12.1 + 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−9.042054518083053428677484530131, −8.339060790661856539246731331594, −7.69114120500579117741102204067, −6.52698655764796225399518207453, −5.92072147675580421453070366489, −5.14954077335541860969919494177, −4.46120449823935171556751709637, −3.64181961349167911041177023764, −2.39793839731160110273187275550, −1.14151916995512427173884168511, 0.06025621402253191250400830490, 1.83140658517299868192466036553, 2.57686774402473848559739337691, 3.69373856823323570745195890226, 4.57922510915999335267638290724, 5.67412978390412511833714221993, 6.05126431839718000420040690428, 7.21524501240285829771186039566, 7.43455749170963606657556499136, 8.231054859885235506651683263536

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