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Properties

Label 2-480-12.11-c3-0-18
Degree $2$
Conductor $480$
Sign $0.251 - 0.967i$
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  + (4.47 − 2.63i)3-s + 5i·5-s + 30.6i·7-s + (13.1 − 23.5i)9-s + 8.44·11-s + 1.42·13-s + (13.1 + 22.3i)15-s + 53.1i·17-s − 42.2i·19-s + (80.8 + 137. i)21-s − 75.5·23-s − 25·25-s + (−3.29 − 140. i)27-s + 184. i·29-s + 313. i·31-s + ⋯
L(s)  = 1  + (0.862 − 0.506i)3-s + 0.447i·5-s + 1.65i·7-s + (0.486 − 0.873i)9-s + 0.231·11-s + 0.0303·13-s + (0.226 + 0.385i)15-s + 0.758i·17-s − 0.510i·19-s + (0.839 + 1.42i)21-s − 0.684·23-s − 0.200·25-s + (−0.0234 − 0.999i)27-s + 1.17i·29-s + 1.81i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.251 - 0.967i$
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: \((2,\ 480,\ (\ :3/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.341916424\)
\(L(\frac12)\) \(\approx\) \(2.341916424\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.47 + 2.63i)T \)
5 \( 1 - 5iT \)
good7 \( 1 - 30.6iT - 343T^{2} \)
11 \( 1 - 8.44T + 1.33e3T^{2} \)
13 \( 1 - 1.42T + 2.19e3T^{2} \)
17 \( 1 - 53.1iT - 4.91e3T^{2} \)
19 \( 1 + 42.2iT - 6.85e3T^{2} \)
23 \( 1 + 75.5T + 1.21e4T^{2} \)
29 \( 1 - 184. iT - 2.43e4T^{2} \)
31 \( 1 - 313. iT - 2.97e4T^{2} \)
37 \( 1 + 344.T + 5.06e4T^{2} \)
41 \( 1 - 349. iT - 6.89e4T^{2} \)
43 \( 1 - 161. iT - 7.95e4T^{2} \)
47 \( 1 - 505.T + 1.03e5T^{2} \)
53 \( 1 + 215. iT - 1.48e5T^{2} \)
59 \( 1 - 582.T + 2.05e5T^{2} \)
61 \( 1 - 635.T + 2.26e5T^{2} \)
67 \( 1 + 143. iT - 3.00e5T^{2} \)
71 \( 1 - 473.T + 3.57e5T^{2} \)
73 \( 1 + 224.T + 3.89e5T^{2} \)
79 \( 1 + 1.18e3iT - 4.93e5T^{2} \)
83 \( 1 - 425.T + 5.71e5T^{2} \)
89 \( 1 - 827. iT - 7.04e5T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−10.73485955758602297125147587278, −9.658516446279654317221140776998, −8.750443167365372381109077642368, −8.351005063512803594257827328694, −7.05689159361413699606934613133, −6.29231560805457687642204855492, −5.18524947955364219698784835617, −3.58618686981811893462097878401, −2.64678799468567549033257261973, −1.63991820468528114887015840515, 0.65483998739696972479473176002, 2.18367607022237370708229853434, 3.83474752257244306009179302812, 4.16314535429901502412125654577, 5.49098550105813514325965193073, 7.02918235162976991260471433097, 7.70345540179599155742580008068, 8.581417976966516017463930155741, 9.658434663840766186054255920559, 10.16431953796721331736841886774

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