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Properties

Label 2-360-120.59-c3-0-28
Degree $2$
Conductor $360$
Sign $-0.632 - 0.774i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.33 + 1.59i)2-s + (2.89 + 7.45i)4-s + (3.30 + 10.6i)5-s + 10.1·7-s + (−5.17 + 22.0i)8-s + (−9.35 + 30.2i)10-s + 13.8i·11-s + 41.4·13-s + (23.7 + 16.2i)14-s + (−47.2 + 43.1i)16-s − 27.7·17-s + 15.2·19-s + (−70.1 + 55.5i)20-s + (−22.1 + 32.3i)22-s − 3.27i·23-s + ⋯
L(s)  = 1  + (0.825 + 0.565i)2-s + (0.361 + 0.932i)4-s + (0.295 + 0.955i)5-s + 0.548·7-s + (−0.228 + 0.973i)8-s + (−0.295 + 0.955i)10-s + 0.380i·11-s + 0.884·13-s + (0.452 + 0.309i)14-s + (−0.738 + 0.674i)16-s − 0.395·17-s + 0.184·19-s + (−0.783 + 0.621i)20-s + (−0.214 + 0.313i)22-s − 0.0296i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.632 - 0.774i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.632 - 0.774i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.265459723\)
\(L(\frac12)\) \(\approx\) \(3.265459723\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.33 - 1.59i)T \)
3 \( 1 \)
5 \( 1 + (-3.30 - 10.6i)T \)
good7 \( 1 - 10.1T + 343T^{2} \)
11 \( 1 - 13.8iT - 1.33e3T^{2} \)
13 \( 1 - 41.4T + 2.19e3T^{2} \)
17 \( 1 + 27.7T + 4.91e3T^{2} \)
19 \( 1 - 15.2T + 6.85e3T^{2} \)
23 \( 1 + 3.27iT - 1.21e4T^{2} \)
29 \( 1 + 100.T + 2.43e4T^{2} \)
31 \( 1 + 8.55iT - 2.97e4T^{2} \)
37 \( 1 - 190.T + 5.06e4T^{2} \)
41 \( 1 - 211. iT - 6.89e4T^{2} \)
43 \( 1 + 398. iT - 7.95e4T^{2} \)
47 \( 1 - 171. iT - 1.03e5T^{2} \)
53 \( 1 - 124. iT - 1.48e5T^{2} \)
59 \( 1 - 354. iT - 2.05e5T^{2} \)
61 \( 1 - 397. iT - 2.26e5T^{2} \)
67 \( 1 + 554. iT - 3.00e5T^{2} \)
71 \( 1 - 961.T + 3.57e5T^{2} \)
73 \( 1 + 920. iT - 3.89e5T^{2} \)
79 \( 1 - 683. iT - 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 - 581. iT - 7.04e5T^{2} \)
97 \( 1 + 1.34e3iT - 9.12e5T^{2} \)
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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

−11.30559522239119136798630975260, −10.78985414936209373893408710865, −9.442079786283692730971161359427, −8.251244300803710963913075716268, −7.36837679190693744775017734988, −6.47353795122963804722676810894, −5.61770443773711010161469027198, −4.39298451048835866221283616169, −3.28270823314350975732157307899, −2.01560298694069163897115269255, 0.885568718994720806417842690805, 2.03115223827986085968032618603, 3.61468969420740718943133854959, 4.67110727038587121479193989788, 5.54633104977779358408797499582, 6.46387013664142914886857454805, 7.975609856736132936729727132749, 8.975803387926905106591451879316, 9.853579302473836088772839260496, 11.02895798965654586114069709395

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