Properties

Label 2-180-20.19-c2-0-9
Degree $2$
Conductor $180$
Sign $-0.0450 - 0.998i$
Analytic cond. $4.90464$
Root an. cond. $2.21464$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.656 + 1.88i)2-s + (−3.13 − 2.48i)4-s + (3.27 + 3.77i)5-s + 9.55·7-s + (6.74 − 4.29i)8-s + (−9.28 + 3.70i)10-s − 9.92i·11-s + 7.55i·13-s + (−6.27 + 18.0i)14-s + (3.68 + 15.5i)16-s + 17.1i·17-s + 26.1i·19-s + (−0.900 − 19.9i)20-s + (18.7 + 6.51i)22-s + 1.67·23-s + ⋯
L(s)  = 1  + (−0.328 + 0.944i)2-s + (−0.784 − 0.620i)4-s + (0.654 + 0.755i)5-s + 1.36·7-s + (0.843 − 0.537i)8-s + (−0.928 + 0.370i)10-s − 0.902i·11-s + 0.581i·13-s + (−0.448 + 1.28i)14-s + (0.230 + 0.973i)16-s + 1.01i·17-s + 1.37i·19-s + (−0.0450 − 0.998i)20-s + (0.852 + 0.296i)22-s + 0.0728·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0450 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $-0.0450 - 0.998i$
Analytic conductor: \(4.90464\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1),\ -0.0450 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01264 + 1.05930i\)
\(L(\frac12)\) \(\approx\) \(1.01264 + 1.05930i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.656 - 1.88i)T \)
3 \( 1 \)
5 \( 1 + (-3.27 - 3.77i)T \)
good7 \( 1 - 9.55T + 49T^{2} \)
11 \( 1 + 9.92iT - 121T^{2} \)
13 \( 1 - 7.55iT - 169T^{2} \)
17 \( 1 - 17.1iT - 289T^{2} \)
19 \( 1 - 26.1iT - 361T^{2} \)
23 \( 1 - 1.67T + 529T^{2} \)
29 \( 1 - 0.350T + 841T^{2} \)
31 \( 1 + 46.0iT - 961T^{2} \)
37 \( 1 + 22.6iT - 1.36e3T^{2} \)
41 \( 1 - 77.2T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + 14.0T + 2.20e3T^{2} \)
53 \( 1 - 22.6iT - 2.80e3T^{2} \)
59 \( 1 + 94.7iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 + 29.8T + 4.48e3T^{2} \)
71 \( 1 + 7.19iT - 5.04e3T^{2} \)
73 \( 1 + 34.3iT - 5.32e3T^{2} \)
79 \( 1 - 46.0iT - 6.24e3T^{2} \)
83 \( 1 + 24.1T + 6.88e3T^{2} \)
89 \( 1 + 100.T + 7.92e3T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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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

−12.99929583059927608695128501640, −11.37117247297276197975845410117, −10.63306437867562367626313373909, −9.567133495900019129078208960628, −8.383833551843684843797012810920, −7.65210848480699609956275395592, −6.28622600679925237112160574794, −5.55356571855635847609809054979, −4.04770244209767630608378260453, −1.71587304703013096225568445561, 1.18023235947412593160093871228, 2.52263218226953585578463943496, 4.61319451729424821952529536832, 5.14189127089276200382354193889, 7.29127173599130659856690748690, 8.411603657082906222746485080446, 9.227550572252362366142971042030, 10.20534653894084308889940183200, 11.22488509535935284925033864634, 12.07326101094153187450003034974

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