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Properties

Label 2-180-36.23-c1-0-15
Degree $2$
Conductor $180$
Sign $0.790 - 0.612i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  + (1.08 + 0.901i)2-s + (1.35 − 1.07i)3-s + (0.373 + 1.96i)4-s + (0.866 + 0.5i)5-s + (2.44 + 0.0473i)6-s + (−3.33 + 1.92i)7-s + (−1.36 + 2.47i)8-s + (0.673 − 2.92i)9-s + (0.492 + 1.32i)10-s + (−1.40 − 2.44i)11-s + (2.62 + 2.26i)12-s + (2.89 − 5.01i)13-s + (−5.36 − 0.909i)14-s + (1.71 − 0.256i)15-s + (−3.72 + 1.46i)16-s + 2.42i·17-s + ⋯
L(s)  = 1  + (0.770 + 0.637i)2-s + (0.782 − 0.622i)3-s + (0.186 + 0.982i)4-s + (0.387 + 0.223i)5-s + (0.999 + 0.0193i)6-s + (−1.25 + 0.727i)7-s + (−0.482 + 0.875i)8-s + (0.224 − 0.974i)9-s + (0.155 + 0.419i)10-s + (−0.424 − 0.736i)11-s + (0.757 + 0.652i)12-s + (0.803 − 1.39i)13-s + (−1.43 − 0.242i)14-s + (0.442 − 0.0661i)15-s + (−0.930 + 0.366i)16-s + 0.588i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87888 + 0.642967i\)
\(L(\frac12)\) \(\approx\) \(1.87888 + 0.642967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.901i)T \)
3 \( 1 + (-1.35 + 1.07i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (3.33 - 1.92i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.40 + 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.89 + 5.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.42iT - 17T^{2} \)
19 \( 1 + 4.07iT - 19T^{2} \)
23 \( 1 + (2.17 - 3.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.07 - 3.50i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.94 - 1.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.29T + 37T^{2} \)
41 \( 1 + (-0.0948 - 0.0547i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.178 + 0.102i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.38 - 9.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.55iT - 53T^{2} \)
59 \( 1 + (-4.16 + 7.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.06 + 5.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + (-4.32 + 2.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 6.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.134iT - 89T^{2} \)
97 \( 1 + (-6.49 - 11.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.15341387010187504838242411220, −12.35358548980942670389793781753, −10.94390716073905727167777984570, −9.418393330402899814532253426402, −8.519419357719347093232542519125, −7.52004881504634829772273416559, −6.25842351288594657943902818371, −5.71494792131493667477696257428, −3.48514170413834105018512543571, −2.76209659592867522295161165518, 2.17958966291193537124086341469, 3.66223747348209957498230313156, 4.45812603562519937326494647715, 6.01428231130464756466266786669, 7.20319960110983224540878470606, 8.958476543518520715400647888592, 9.860859845518024845486860801010, 10.30336307713309906224764240123, 11.62528890485667078076973382771, 12.88307210462866962988904100599

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