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Properties

Label 2-2970-99.32-c1-0-24
Degree $2$
Conductor $2970$
Sign $-0.751 + 0.659i$
Analytic cond. $23.7155$
Root an. cond. $4.86986$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + (−3.44 + 1.98i)7-s + 0.999·8-s + 0.999i·10-s + (−1.94 + 2.68i)11-s + (−1.91 − 1.10i)13-s + (3.44 + 1.98i)14-s + (−0.5 − 0.866i)16-s + 2.92·17-s + 5.09i·19-s + (0.866 − 0.499i)20-s + (3.29 + 0.340i)22-s + (3.45 + 1.99i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−1.30 + 0.751i)7-s + 0.353·8-s + 0.316i·10-s + (−0.586 + 0.810i)11-s + (−0.531 − 0.306i)13-s + (0.920 + 0.531i)14-s + (−0.125 − 0.216i)16-s + 0.709·17-s + 1.16i·19-s + (0.193 − 0.111i)20-s + (0.703 + 0.0726i)22-s + (0.720 + 0.416i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (2771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ -0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2658263236\)
\(L(\frac12)\) \(\approx\) \(0.2658263236\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (1.94 - 2.68i)T \)
good7 \( 1 + (3.44 - 1.98i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (1.91 + 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 - 5.09iT - 19T^{2} \)
23 \( 1 + (-3.45 - 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.94 + 3.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.189 - 0.328i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.03T + 37T^{2} \)
41 \( 1 + (2.82 - 4.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.34 - 3.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.07 + 4.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.04iT - 53T^{2} \)
59 \( 1 + (11.5 + 6.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.64 - 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.41 - 4.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + 1.49iT - 73T^{2} \)
79 \( 1 + (9.26 - 5.35i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.29 - 5.70i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 + (6.50 + 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.503030745704765751430416129799, −7.79662017360651733302945060034, −7.15847689934722652779329733478, −6.10525411109902927147807525118, −5.36104281035294838121705781723, −4.41992144588403060005611663682, −3.35598291814778965982943739992, −2.82040235684697125848774929936, −1.68596566882577506335683654480, −0.12541235074619010561245385226, 0.857917818779003390025969451599, 2.70369946685520305955147501726, 3.40124105928022846597938662608, 4.39635856180805153628460607637, 5.32900880756322604029860223297, 6.17261854448842205882698674748, 6.98809216445469620309854966848, 7.30873530167880490580601191364, 8.197056155623910412256413851174, 9.093296450880709538384271176291

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