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Properties

Label 2-360-5.4-c3-0-3
Degree $2$
Conductor $360$
Sign $-0.999 + 0.0160i$
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  + (0.178 + 11.1i)5-s + 35.0i·7-s − 25.6·11-s + 37.6i·13-s − 95.7i·17-s − 50.8·19-s − 110. i·23-s + (−124. + 4i)25-s − 54.5·29-s + 198.·31-s + (−392. + 6.27i)35-s − 266. i·37-s − 103.·41-s + 108i·43-s + 597. i·47-s + ⋯
L(s)  = 1  + (0.0160 + 0.999i)5-s + 1.89i·7-s − 0.702·11-s + 0.803i·13-s − 1.36i·17-s − 0.614·19-s − 1.00i·23-s + (−0.999 + 0.0320i)25-s − 0.349·29-s + 1.14·31-s + (−1.89 + 0.0303i)35-s − 1.18i·37-s − 0.395·41-s + 0.383i·43-s + 1.85i·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.0160i$
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: \((2,\ 360,\ (\ :3/2),\ -0.999 + 0.0160i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9206442644\)
\(L(\frac12)\) \(\approx\) \(0.9206442644\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.178 - 11.1i)T \)
good7 \( 1 - 35.0iT - 343T^{2} \)
11 \( 1 + 25.6T + 1.33e3T^{2} \)
13 \( 1 - 37.6iT - 2.19e3T^{2} \)
17 \( 1 + 95.7iT - 4.91e3T^{2} \)
19 \( 1 + 50.8T + 6.85e3T^{2} \)
23 \( 1 + 110. iT - 1.21e4T^{2} \)
29 \( 1 + 54.5T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 + 266. iT - 5.06e4T^{2} \)
41 \( 1 + 103.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 - 597. iT - 1.03e5T^{2} \)
53 \( 1 - 305. iT - 1.48e5T^{2} \)
59 \( 1 + 223.T + 2.05e5T^{2} \)
61 \( 1 - 485.T + 2.26e5T^{2} \)
67 \( 1 - 876. iT - 3.00e5T^{2} \)
71 \( 1 + 585.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 + 685.T + 4.93e5T^{2} \)
83 \( 1 + 305. iT - 5.71e5T^{2} \)
89 \( 1 - 887.T + 7.04e5T^{2} \)
97 \( 1 - 556. iT - 9.12e5T^{2} \)
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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

−11.55849871114377940744168934483, −10.64275028147129729567978165398, −9.536756266007061111307207361249, −8.815545948421127036090702956684, −7.72324763506225258568545653863, −6.57737052942814156783056185796, −5.79005574701098173494283899136, −4.63450071681656360364176290209, −2.85672941268375815133128365073, −2.29606688338084363464212646172, 0.31308427072404656831704842922, 1.51659396911693286692796202975, 3.54695242992790782613773384066, 4.44717054520076828041690044270, 5.51366433753322801050170142774, 6.80456721545714798075492248635, 7.915416067979764542472286988502, 8.404451159977627422864634050446, 9.972922254195106601572091403021, 10.33176899079572963513570766749

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