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Properties

Label 2-480-12.11-c3-0-17
Degree $2$
Conductor $480$
Sign $-0.132 - 0.991i$
Analytic cond. $28.3209$
Root an. cond. $5.32174$
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  + (4.12 + 3.15i)3-s − 5i·5-s + 3.91i·7-s + (7.07 + 26.0i)9-s − 0.0664·11-s − 14.4·13-s + (15.7 − 20.6i)15-s + 99.3i·17-s + 10.8i·19-s + (−12.3 + 16.1i)21-s + 44.8·23-s − 25·25-s + (−53.0 + 129. i)27-s + 134. i·29-s + 10.0i·31-s + ⋯
L(s)  = 1  + (0.794 + 0.607i)3-s − 0.447i·5-s + 0.211i·7-s + (0.262 + 0.965i)9-s − 0.00182·11-s − 0.309·13-s + (0.271 − 0.355i)15-s + 1.41i·17-s + 0.130i·19-s + (−0.128 + 0.167i)21-s + 0.406·23-s − 0.200·25-s + (−0.378 + 0.925i)27-s + 0.860i·29-s + 0.0580i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $-0.132 - 0.991i$
Analytic conductor: \(28.3209\)
Root analytic conductor: \(5.32174\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :3/2),\ -0.132 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.210322283\)
\(L(\frac12)\) \(\approx\) \(2.210322283\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.12 - 3.15i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 3.91iT - 343T^{2} \)
11 \( 1 + 0.0664T + 1.33e3T^{2} \)
13 \( 1 + 14.4T + 2.19e3T^{2} \)
17 \( 1 - 99.3iT - 4.91e3T^{2} \)
19 \( 1 - 10.8iT - 6.85e3T^{2} \)
23 \( 1 - 44.8T + 1.21e4T^{2} \)
29 \( 1 - 134. iT - 2.43e4T^{2} \)
31 \( 1 - 10.0iT - 2.97e4T^{2} \)
37 \( 1 - 152.T + 5.06e4T^{2} \)
41 \( 1 - 267. iT - 6.89e4T^{2} \)
43 \( 1 - 252. iT - 7.95e4T^{2} \)
47 \( 1 + 87.1T + 1.03e5T^{2} \)
53 \( 1 + 28.9iT - 1.48e5T^{2} \)
59 \( 1 + 280.T + 2.05e5T^{2} \)
61 \( 1 - 702.T + 2.26e5T^{2} \)
67 \( 1 - 180. iT - 3.00e5T^{2} \)
71 \( 1 + 927.T + 3.57e5T^{2} \)
73 \( 1 + 516.T + 3.89e5T^{2} \)
79 \( 1 + 787. iT - 4.93e5T^{2} \)
83 \( 1 + 1.13e3T + 5.71e5T^{2} \)
89 \( 1 - 649. iT - 7.04e5T^{2} \)
97 \( 1 - 796.T + 9.12e5T^{2} \)
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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

−10.61232322875240670243792791087, −9.879122919053269339381244986054, −8.953849593307443497342395214237, −8.330976646203553419822050051434, −7.41244133661694304980997287367, −6.05272486205360316098642214254, −4.93516648961412805118073072655, −4.02309150513693987734522255632, −2.88077562756717137675757898937, −1.55510983798189033734594980044, 0.62910363730288844618696197222, 2.22362090687669637018395496685, 3.13540758469047133572236603814, 4.36132237752516279234485832568, 5.76884145839149262575698635042, 7.01222799478001481675888110173, 7.39832562858251680009520932742, 8.516526405292275252249814020628, 9.393703258902445253784810214149, 10.17058989871323064331255790478

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