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Properties

Label 2-360-5.4-c3-0-14
Degree $2$
Conductor $360$
Sign $0.0160 + 0.999i$
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  + (−11.1 + 0.178i)5-s + 33.0i·7-s − 48.3·11-s − 60.3i·13-s − 17.7i·17-s + 130.·19-s − 70.8i·23-s + (124. − 4i)25-s + 104.·29-s − 210.·31-s + (−5.91 − 369. i)35-s − 300. i·37-s − 240.·41-s − 108i·43-s − 278. i·47-s + ⋯
L(s)  = 1  + (−0.999 + 0.0160i)5-s + 1.78i·7-s − 1.32·11-s − 1.28i·13-s − 0.253i·17-s + 1.58·19-s − 0.642i·23-s + (0.999 − 0.0320i)25-s + 0.669·29-s − 1.21·31-s + (−0.0285 − 1.78i)35-s − 1.33i·37-s − 0.914·41-s − 0.383i·43-s − 0.865i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0160 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0160 + 0.999i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ 0.0160 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7021546860\)
\(L(\frac12)\) \(\approx\) \(0.7021546860\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 - 0.178i)T \)
good7 \( 1 - 33.0iT - 343T^{2} \)
11 \( 1 + 48.3T + 1.33e3T^{2} \)
13 \( 1 + 60.3iT - 2.19e3T^{2} \)
17 \( 1 + 17.7iT - 4.91e3T^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 + 70.8iT - 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 + 210.T + 2.97e4T^{2} \)
37 \( 1 + 300. iT - 5.06e4T^{2} \)
41 \( 1 + 240.T + 6.89e4T^{2} \)
43 \( 1 + 108iT - 7.95e4T^{2} \)
47 \( 1 + 278. iT - 1.03e5T^{2} \)
53 \( 1 + 328. iT - 1.48e5T^{2} \)
59 \( 1 - 889.T + 2.05e5T^{2} \)
61 \( 1 + 241.T + 2.26e5T^{2} \)
67 \( 1 + 103. iT - 3.00e5T^{2} \)
71 \( 1 - 277.T + 3.57e5T^{2} \)
73 \( 1 - 274. iT - 3.89e5T^{2} \)
79 \( 1 + 366.T + 4.93e5T^{2} \)
83 \( 1 + 57.7iT - 5.71e5T^{2} \)
89 \( 1 + 203.T + 7.04e5T^{2} \)
97 \( 1 + 1.28e3iT - 9.12e5T^{2} \)
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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

−10.90352420818022613082000748899, −9.883099856869820687307331165954, −8.700691978802399485378716724152, −8.097887574638478178755252473662, −7.17182853085261072349251268433, −5.53057477596573220121662938385, −5.18271252761309206008955252382, −3.34728210724607003680725586190, −2.49828381229420339509277005119, −0.27663243875411126014294360742, 1.16675965794678730121946221020, 3.21625457937593366919834296702, 4.16598643991635338589664171839, 5.10262990138862758897610731753, 6.83272327907529629326110906401, 7.46148106142921783043964068591, 8.156443416318805138056098115540, 9.542197334303413333543765491428, 10.44773813930500117743002433673, 11.20011376813218585601221880611

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