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Properties

Label 2-180-180.103-c1-0-15
Degree $2$
Conductor $180$
Sign $0.739 + 0.672i$
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  + (−0.608 − 1.27i)2-s + (1.70 − 0.286i)3-s + (−1.25 + 1.55i)4-s + (1.09 + 1.94i)5-s + (−1.40 − 2.00i)6-s + (0.0500 − 0.186i)7-s + (2.74 + 0.662i)8-s + (2.83 − 0.979i)9-s + (1.81 − 2.58i)10-s + (4.51 − 2.60i)11-s + (−1.70 + 3.01i)12-s + (−6.23 + 1.67i)13-s + (−0.269 + 0.0497i)14-s + (2.43 + 3.01i)15-s + (−0.827 − 3.91i)16-s + (−0.305 + 0.305i)17-s + ⋯
L(s)  = 1  + (−0.430 − 0.902i)2-s + (0.986 − 0.165i)3-s + (−0.629 + 0.776i)4-s + (0.491 + 0.871i)5-s + (−0.573 − 0.819i)6-s + (0.0189 − 0.0706i)7-s + (0.972 + 0.234i)8-s + (0.945 − 0.326i)9-s + (0.575 − 0.818i)10-s + (1.36 − 0.785i)11-s + (−0.492 + 0.870i)12-s + (−1.72 + 0.463i)13-s + (−0.0718 + 0.0133i)14-s + (0.628 + 0.777i)15-s + (−0.206 − 0.978i)16-s + (−0.0740 + 0.0740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(180\)    =    \(2^{2} \cdot 3^{2} \cdot 5\)
Sign: $0.739 + 0.672i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{180} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 180,\ (\ :1/2),\ 0.739 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21429 - 0.469559i\)
\(L(\frac12)\) \(\approx\) \(1.21429 - 0.469559i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 + 1.27i)T \)
3 \( 1 + (-1.70 + 0.286i)T \)
5 \( 1 + (-1.09 - 1.94i)T \)
good7 \( 1 + (-0.0500 + 0.186i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.51 + 2.60i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.23 - 1.67i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.305 - 0.305i)T - 17iT^{2} \)
19 \( 1 + 2.39T + 19T^{2} \)
23 \( 1 + (1.13 + 4.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.28 - 1.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.124 + 0.0717i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.83 + 2.83i)T - 37iT^{2} \)
41 \( 1 + (1.76 - 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.96 + 2.13i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.07 - 7.73i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.86 + 3.86i)T + 53iT^{2} \)
59 \( 1 + (-0.587 + 1.01i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.54 - 4.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.12 - 2.44i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.80iT - 71T^{2} \)
73 \( 1 + (1.96 + 1.96i)T + 73iT^{2} \)
79 \( 1 + (1.89 + 3.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.57 - 2.03i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 13.0iT - 89T^{2} \)
97 \( 1 + (5.44 + 1.45i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.47019828687867227431965314709, −11.53424244529065122518245154566, −10.38403910371957560248403803312, −9.555028569667480129821876614174, −8.820833040241880288026984340867, −7.56391052317291490959942744325, −6.57673531642388954159474865302, −4.34706779023815450139167258986, −3.10332876546588907522857385488, −1.96439215768856286146624885039, 1.84980987892155339977918626832, 4.22762682582232009471030365738, 5.20583931634471760248760414831, 6.77196056123062974622941435804, 7.74529090911676008652954564890, 8.787652851536677291772616519486, 9.598641851597359377628034895789, 10.04073386007532197743499289994, 12.05942540428267100547003392912, 13.05086924768586240215355170737

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