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Properties

Label 2-480-15.8-c1-0-11
Degree $2$
Conductor $480$
Sign $0.824 + 0.565i$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 + 0.391i)3-s + (1.95 − 1.09i)5-s + (0.246 − 0.246i)7-s + (2.69 − 1.31i)9-s + 1.86i·11-s + (−3.25 − 3.25i)13-s + (−2.86 + 2.60i)15-s + (2.34 + 2.34i)17-s − 5.02i·19-s + (−0.319 + 0.512i)21-s + (3.94 − 3.94i)23-s + (2.61 − 4.25i)25-s + (−4.03 + 3.28i)27-s + 7.57·29-s + 10.7·31-s + ⋯
L(s)  = 1  + (−0.974 + 0.225i)3-s + (0.872 − 0.487i)5-s + (0.0931 − 0.0931i)7-s + (0.898 − 0.439i)9-s + 0.562i·11-s + (−0.903 − 0.903i)13-s + (−0.740 + 0.672i)15-s + (0.568 + 0.568i)17-s − 1.15i·19-s + (−0.0697 + 0.111i)21-s + (0.822 − 0.822i)23-s + (0.523 − 0.851i)25-s + (−0.775 + 0.631i)27-s + 1.40·29-s + 1.92·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.824 + 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14739 - 0.355930i\)
\(L(\frac12)\) \(\approx\) \(1.14739 - 0.355930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.391i)T \)
5 \( 1 + (-1.95 + 1.09i)T \)
good7 \( 1 + (-0.246 + 0.246i)T - 7iT^{2} \)
11 \( 1 - 1.86iT - 11T^{2} \)
13 \( 1 + (3.25 + 3.25i)T + 13iT^{2} \)
17 \( 1 + (-2.34 - 2.34i)T + 17iT^{2} \)
19 \( 1 + 5.02iT - 19T^{2} \)
23 \( 1 + (-3.94 + 3.94i)T - 23iT^{2} \)
29 \( 1 - 7.57T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (0.619 - 0.619i)T - 37iT^{2} \)
41 \( 1 + 3.66iT - 41T^{2} \)
43 \( 1 + (3.73 + 3.73i)T + 43iT^{2} \)
47 \( 1 + (1.11 + 1.11i)T + 47iT^{2} \)
53 \( 1 + (2.80 - 2.80i)T - 53iT^{2} \)
59 \( 1 + 7.94T + 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 + (-6.81 + 6.81i)T - 67iT^{2} \)
71 \( 1 - 9.81iT - 71T^{2} \)
73 \( 1 + (6.23 + 6.23i)T + 73iT^{2} \)
79 \( 1 + 9.09iT - 79T^{2} \)
83 \( 1 + (5.27 - 5.27i)T - 83iT^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 + (11.5 - 11.5i)T - 97iT^{2} \)
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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

−10.62226320719374310323052682013, −10.20015686922984594521838076857, −9.378039873946219716753379338915, −8.243136855157160290683950638594, −6.98965435827885545660322800409, −6.18863192993480475932660969714, −5.05761240751040257314003301784, −4.62248230478567928446801655817, −2.69097646881341577720414632237, −0.957835349918271323016497362163, 1.43281155557151458823021171897, 2.91239239581284041977542574729, 4.63186370201933397111478810284, 5.52570408426897739334416464611, 6.42182794421574425519974613234, 7.12539883888158466348530928547, 8.297763655667165845723945021886, 9.779845249104119311387685904241, 10.01182659704459274508038003390, 11.20685479372593832264566898388

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