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Properties

Label 2-2970-99.32-c1-0-45
Degree $2$
Conductor $2970$
Sign $-0.691 - 0.722i$
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 + (1.93 − 1.11i)7-s + 0.999·8-s + 0.999i·10-s + (−3.16 + 0.989i)11-s + (2.70 + 1.55i)13-s + (−1.93 − 1.11i)14-s + (−0.5 − 0.866i)16-s − 3.53·17-s + 0.649i·19-s + (0.866 − 0.499i)20-s + (2.43 + 2.24i)22-s + (−1.72 − 0.997i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.732 − 0.422i)7-s + 0.353·8-s + 0.316i·10-s + (−0.954 + 0.298i)11-s + (0.749 + 0.432i)13-s + (−0.517 − 0.299i)14-s + (−0.125 − 0.216i)16-s − 0.856·17-s + 0.148i·19-s + (0.193 − 0.111i)20-s + (0.520 + 0.478i)22-s + (−0.360 − 0.208i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2970\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 11\)
Sign: $-0.691 - 0.722i$
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: \((2,\ 2970,\ (\ :1/2),\ -0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03060105221\)
\(L(\frac12)\) \(\approx\) \(0.03060105221\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 - 0.989i)T \)
good7 \( 1 + (-1.93 + 1.11i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-2.70 - 1.55i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.53T + 17T^{2} \)
19 \( 1 - 0.649iT - 19T^{2} \)
23 \( 1 + (1.72 + 0.997i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0531 - 0.0920i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.17 + 7.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.00T + 37T^{2} \)
41 \( 1 + (-1.38 + 2.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.0 - 5.78i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.60 - 3.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.95iT - 53T^{2} \)
59 \( 1 + (-2.80 - 1.61i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.4 - 6.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.16iT - 71T^{2} \)
73 \( 1 + 1.40iT - 73T^{2} \)
79 \( 1 + (2.59 - 1.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.91 + 8.51i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.269iT - 89T^{2} \)
97 \( 1 + (2.53 + 4.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.185277946046063376390842177642, −7.86905612743026942269792958866, −6.92547547566112702503697245916, −5.99085947666554262876438294667, −4.78960929742909969879054362858, −4.39801202405187856337639342854, −3.40753735051644688051977820190, −2.30520203169426844855296907307, −1.38952638685354723314157042762, −0.01104611806127722097041365899, 1.50998851648342910718370617780, 2.69455157020382355438871919787, 3.72229242693927240393326091096, 4.86772970638141800711174538792, 5.32867769684133882758897259462, 6.30906875381173244214576991949, 6.96795815596051574622572900794, 7.915559029896795994757541472574, 8.380598616875817552614921621715, 8.829318483121928855393305205802

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