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Properties

Label 2-2970-99.32-c1-0-4
Degree $2$
Conductor $2970$
Sign $0.296 - 0.954i$
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 + (−1.01 + 0.585i)7-s + 0.999·8-s + 0.999i·10-s + (0.345 + 3.29i)11-s + (−4.36 − 2.52i)13-s + (1.01 + 0.585i)14-s + (−0.5 − 0.866i)16-s + 3.55·17-s − 6.12i·19-s + (0.866 − 0.499i)20-s + (2.68 − 1.94i)22-s + (0.456 + 0.263i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (−0.383 + 0.221i)7-s + 0.353·8-s + 0.316i·10-s + (0.104 + 0.994i)11-s + (−1.21 − 0.699i)13-s + (0.271 + 0.156i)14-s + (−0.125 − 0.216i)16-s + 0.861·17-s − 1.40i·19-s + (0.193 − 0.111i)20-s + (0.572 − 0.415i)22-s + (0.0950 + 0.0549i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $0.296 - 0.954i$
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: \((2,\ 2970,\ (\ :1/2),\ 0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5602209301\)
\(L(\frac12)\) \(\approx\) \(0.5602209301\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.345 - 3.29i)T \)
good7 \( 1 + (1.01 - 0.585i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (4.36 + 2.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + 6.12iT - 19T^{2} \)
23 \( 1 + (-0.456 - 0.263i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.386 + 0.670i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 4.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.26T + 37T^{2} \)
41 \( 1 + (1.05 - 1.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.762 + 0.440i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.56 - 3.78i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.36iT - 53T^{2} \)
59 \( 1 + (-10.8 - 6.28i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.48 - 2.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.70 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 + 4.47iT - 73T^{2} \)
79 \( 1 + (5.60 - 3.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.22 + 12.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + (2.49 + 4.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.067697252959140572059134304680, −8.149646486581559174982702956546, −7.47922084571660049338815086594, −6.92093258811029644191323079521, −5.71670693632795863934368686841, −4.82794945080328752789151143615, −4.21489920740228987944322862163, −2.98802906958526622316418291305, −2.45275931707042244963393386219, −1.03668027149828637820733498788, 0.23407948842954465895156279485, 1.64345516672139482482987158319, 3.05584105241563406815642053765, 3.80447721949718724568830964605, 4.83656697818331908899792073334, 5.65187527545156569945328776983, 6.46852505655065363757523980514, 7.07678193510968766671753751801, 7.904532710441110062963716781378, 8.370084417514777960374438968290

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