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Properties

Label 2-360-15.8-c3-0-3
Degree $2$
Conductor $360$
Sign $-0.480 - 0.877i$
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  + (−7.08 + 8.64i)5-s + (3.01 − 3.01i)7-s − 64.6i·11-s + (46.4 + 46.4i)13-s + (51.1 + 51.1i)17-s + 75.4i·19-s + (−131. + 131. i)23-s + (−24.6 − 122. i)25-s − 279.·29-s − 7.69·31-s + (4.72 + 47.5i)35-s + (−152. + 152. i)37-s + 295. i·41-s + (200. + 200. i)43-s + (−292. − 292. i)47-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)5-s + (0.163 − 0.163i)7-s − 1.77i·11-s + (0.992 + 0.992i)13-s + (0.730 + 0.730i)17-s + 0.911i·19-s + (−1.19 + 1.19i)23-s + (−0.196 − 0.980i)25-s − 1.78·29-s − 0.0445·31-s + (0.0228 + 0.229i)35-s + (−0.678 + 0.678i)37-s + 1.12i·41-s + (0.710 + 0.710i)43-s + (−0.907 − 0.907i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.480 - 0.877i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.480 - 0.877i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.074087705\)
\(L(\frac12)\) \(\approx\) \(1.074087705\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (7.08 - 8.64i)T \)
good7 \( 1 + (-3.01 + 3.01i)T - 343iT^{2} \)
11 \( 1 + 64.6iT - 1.33e3T^{2} \)
13 \( 1 + (-46.4 - 46.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (-51.1 - 51.1i)T + 4.91e3iT^{2} \)
19 \( 1 - 75.4iT - 6.85e3T^{2} \)
23 \( 1 + (131. - 131. i)T - 1.21e4iT^{2} \)
29 \( 1 + 279.T + 2.43e4T^{2} \)
31 \( 1 + 7.69T + 2.97e4T^{2} \)
37 \( 1 + (152. - 152. i)T - 5.06e4iT^{2} \)
41 \( 1 - 295. iT - 6.89e4T^{2} \)
43 \( 1 + (-200. - 200. i)T + 7.95e4iT^{2} \)
47 \( 1 + (292. + 292. i)T + 1.03e5iT^{2} \)
53 \( 1 + (212. - 212. i)T - 1.48e5iT^{2} \)
59 \( 1 + 55.6T + 2.05e5T^{2} \)
61 \( 1 - 19.2T + 2.26e5T^{2} \)
67 \( 1 + (-437. + 437. i)T - 3.00e5iT^{2} \)
71 \( 1 - 582. iT - 3.57e5T^{2} \)
73 \( 1 + (172. + 172. i)T + 3.89e5iT^{2} \)
79 \( 1 - 767. iT - 4.93e5T^{2} \)
83 \( 1 + (-369. + 369. i)T - 5.71e5iT^{2} \)
89 \( 1 - 394.T + 7.04e5T^{2} \)
97 \( 1 + (291. - 291. i)T - 9.12e5iT^{2} \)
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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

−11.30280207091846633208830039004, −10.65134215951136505343434211013, −9.498144221336004029887619425035, −8.280671328786790633535521250052, −7.77145432253395873453379381751, −6.37895371652999454627907021396, −5.75594916142004425471827716725, −3.87805787783619885301149181654, −3.41099356271653098110444985021, −1.49656815164524061320103505552, 0.38436512624979855600143019174, 1.98529662302810052589802718871, 3.66928289639470738223035424319, 4.72101251344881296527629648789, 5.62542505747591778848919516700, 7.12714974057478889152223865027, 7.85896077318672755315086188889, 8.837728898586267877477508341284, 9.721664733791871697666499226180, 10.73806671334384533963576421934

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