↑

Properties

Label 2-360-15.8-c3-0-2
Degree $2$
Conductor $360$
Sign $-0.995 + 0.0922i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.12 + 9.93i)5-s + (−14.2 + 14.2i)7-s + 38.4i·11-s + (−11.0 − 11.0i)13-s + (−95.0 − 95.0i)17-s + 17.3i·19-s + (115. − 115. i)23-s + (−72.4 + 101. i)25-s − 193.·29-s − 107.·31-s + (−214. − 68.5i)35-s + (43.7 − 43.7i)37-s − 225. i·41-s + (306. + 306. i)43-s + (−220. − 220. i)47-s + ⋯
L(s)  = 1  + (0.458 + 0.888i)5-s + (−0.769 + 0.769i)7-s + 1.05i·11-s + (−0.235 − 0.235i)13-s + (−1.35 − 1.35i)17-s + 0.209i·19-s + (1.04 − 1.04i)23-s + (−0.579 + 0.814i)25-s − 1.24·29-s − 0.620·31-s + (−1.03 − 0.331i)35-s + (0.194 − 0.194i)37-s − 0.858i·41-s + (1.08 + 1.08i)43-s + (−0.684 − 0.684i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.995 + 0.0922i$
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: \((2,\ 360,\ (\ :3/2),\ -0.995 + 0.0922i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5569639297\)
\(L(\frac12)\) \(\approx\) \(0.5569639297\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.12 - 9.93i)T \)
good7 \( 1 + (14.2 - 14.2i)T - 343iT^{2} \)
11 \( 1 - 38.4iT - 1.33e3T^{2} \)
13 \( 1 + (11.0 + 11.0i)T + 2.19e3iT^{2} \)
17 \( 1 + (95.0 + 95.0i)T + 4.91e3iT^{2} \)
19 \( 1 - 17.3iT - 6.85e3T^{2} \)
23 \( 1 + (-115. + 115. i)T - 1.21e4iT^{2} \)
29 \( 1 + 193.T + 2.43e4T^{2} \)
31 \( 1 + 107.T + 2.97e4T^{2} \)
37 \( 1 + (-43.7 + 43.7i)T - 5.06e4iT^{2} \)
41 \( 1 + 225. iT - 6.89e4T^{2} \)
43 \( 1 + (-306. - 306. i)T + 7.95e4iT^{2} \)
47 \( 1 + (220. + 220. i)T + 1.03e5iT^{2} \)
53 \( 1 + (257. - 257. i)T - 1.48e5iT^{2} \)
59 \( 1 + 124.T + 2.05e5T^{2} \)
61 \( 1 + 862.T + 2.26e5T^{2} \)
67 \( 1 + (557. - 557. i)T - 3.00e5iT^{2} \)
71 \( 1 + 246. iT - 3.57e5T^{2} \)
73 \( 1 + (-77.9 - 77.9i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.27e3iT - 4.93e5T^{2} \)
83 \( 1 + (-502. + 502. i)T - 5.71e5iT^{2} \)
89 \( 1 + 958.T + 7.04e5T^{2} \)
97 \( 1 + (-661. + 661. i)T - 9.12e5iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.36841332265430455745714786361, −10.58537362060527284267393742551, −9.471791855581143032636792142587, −9.119038831729163678609171646286, −7.42756048503964122563293665950, −6.78689054658424486304065253827, −5.79530561683643125412480878396, −4.59582528572000506295546261829, −2.98869531204609194643081015006, −2.19589966553866214712575791394, 0.18124043086336825833957873445, 1.64256060782455956033257805963, 3.38090280694812707292876923513, 4.45772242118516664716730517812, 5.72135574858483758530110842494, 6.56087766419437513873147760663, 7.75093702172628306399751906742, 8.919365802507726300846623482270, 9.408981404976625362307316664444, 10.62011599490361583813017472182

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