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Properties

Label 2-2940-35.13-c1-0-8
Degree $2$
Conductor $2940$
Sign $-0.910 - 0.413i$
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 + (−0.523 + 2.17i)5-s − 1.00i·9-s − 0.0458·11-s + (0.0340 − 0.0340i)13-s + (−1.16 − 1.90i)15-s + (3.13 + 3.13i)17-s + 1.19·19-s + (2.02 + 2.02i)23-s + (−4.45 − 2.27i)25-s + (0.707 + 0.707i)27-s − 0.124i·29-s + 9.44i·31-s + (0.0324 − 0.0324i)33-s + (−1.29 + 1.29i)37-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.234 + 0.972i)5-s − 0.333i·9-s − 0.0138·11-s + (0.00945 − 0.00945i)13-s + (−0.301 − 0.492i)15-s + (0.760 + 0.760i)17-s + 0.274·19-s + (0.421 + 0.421i)23-s + (−0.890 − 0.455i)25-s + (0.136 + 0.136i)27-s − 0.0230i·29-s + 1.69i·31-s + (0.00564 − 0.00564i)33-s + (−0.213 + 0.213i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.910 - 0.413i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ -0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.037652493\)
\(L(\frac12)\) \(\approx\) \(1.037652493\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.523 - 2.17i)T \)
7 \( 1 \)
good11 \( 1 + 0.0458T + 11T^{2} \)
13 \( 1 + (-0.0340 + 0.0340i)T - 13iT^{2} \)
17 \( 1 + (-3.13 - 3.13i)T + 17iT^{2} \)
19 \( 1 - 1.19T + 19T^{2} \)
23 \( 1 + (-2.02 - 2.02i)T + 23iT^{2} \)
29 \( 1 + 0.124iT - 29T^{2} \)
31 \( 1 - 9.44iT - 31T^{2} \)
37 \( 1 + (1.29 - 1.29i)T - 37iT^{2} \)
41 \( 1 + 7.53iT - 41T^{2} \)
43 \( 1 + (3.37 + 3.37i)T + 43iT^{2} \)
47 \( 1 + (-7.52 - 7.52i)T + 47iT^{2} \)
53 \( 1 + (-2.27 - 2.27i)T + 53iT^{2} \)
59 \( 1 - 2.35T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 + (-0.762 + 0.762i)T - 67iT^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (9.05 - 9.05i)T - 73iT^{2} \)
79 \( 1 - 2.94iT - 79T^{2} \)
83 \( 1 + (-7.74 + 7.74i)T - 83iT^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (5.38 + 5.38i)T + 97iT^{2} \)
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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

−9.082810192251809838556690232388, −8.355492847663296418826085599820, −7.36094510626890428269669844202, −6.92828861167230086580983495022, −5.92840014877435812806865096662, −5.39249714641884414006150135522, −4.27686143971281668540767905400, −3.51307503115940052221699008432, −2.75527496298251126753797369818, −1.35659153851700149432073088869, 0.38053607568496107962712996082, 1.35916674904791253890525200285, 2.56477110806310466829278345753, 3.73618472918394812541502570655, 4.67307161701963746611411016102, 5.32557762507510282561751572277, 6.03743267693949827341335338967, 6.99214983615082489575265435039, 7.74342352444636964088955287628, 8.282842285330759512991390584569

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