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Properties

Label 2-2970-99.65-c1-0-17
Degree $2$
Conductor $2970$
Sign $-0.277 - 0.960i$
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 + (1.10 + 0.637i)7-s + 0.999·8-s − 0.999i·10-s + (2.15 + 2.51i)11-s + (−1.12 + 0.651i)13-s + (−1.10 + 0.637i)14-s + (−0.5 + 0.866i)16-s + 3.37·17-s − 5.39i·19-s + (0.866 + 0.499i)20-s + (−3.25 + 0.611i)22-s + (−1.82 + 1.05i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.387 + 0.223i)5-s + (0.417 + 0.240i)7-s + 0.353·8-s − 0.316i·10-s + (0.650 + 0.759i)11-s + (−0.313 + 0.180i)13-s + (−0.295 + 0.170i)14-s + (−0.125 + 0.216i)16-s + 0.818·17-s − 1.23i·19-s + (0.193 + 0.111i)20-s + (−0.695 + 0.130i)22-s + (−0.381 + 0.220i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(23.7155\)
Root analytic conductor: \(4.86986\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2970} (791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2970,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.357847771\)
\(L(\frac12)\) \(\approx\) \(1.357847771\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.15 - 2.51i)T \)
good7 \( 1 + (-1.10 - 0.637i)T + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (1.12 - 0.651i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.37T + 17T^{2} \)
19 \( 1 + 5.39iT - 19T^{2} \)
23 \( 1 + (1.82 - 1.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.37 - 4.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.73 - 3.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 + (-4.86 - 8.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.29 + 4.79i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.51 - 3.76i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.11iT - 53T^{2} \)
59 \( 1 + (-2.49 + 1.44i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 2.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.12 - 5.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 14.1iT - 73T^{2} \)
79 \( 1 + (-5.22 - 3.01i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.809 - 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.53iT - 89T^{2} \)
97 \( 1 + (-5.59 + 9.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.014557327907510808639559293679, −8.035601728933992894298371540102, −7.51833714704997544003570913963, −6.80612369647990736336747825629, −6.10649395335543626419062542002, −5.03599501118875268692842163357, −4.53136412600473043331574634415, −3.44208964262719547148922936510, −2.26740561147183655876423355957, −1.07622900253065210621014271652, 0.59155032585783790288069832178, 1.59359972122749886705462359719, 2.77795790607107163392785851283, 3.81225051171735793754693016021, 4.26239955799107916289663427670, 5.46399845958673910856414764456, 6.14430152497880994264668567068, 7.30208446702329889241409344634, 8.016980796033853568087208973320, 8.372606188262293210664626286216

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