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Properties

Label 2-480-40.29-c1-0-8
Degree $2$
Conductor $480$
Sign $-0.367 + 0.930i$
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  − 3-s + (−1.86 + 1.23i)5-s + 0.746i·7-s + 9-s − 5.36i·11-s − 2.92·13-s + (1.86 − 1.23i)15-s − 2.13i·17-s + 1.73i·19-s − 0.746i·21-s − 7.49i·23-s + (1.92 − 4.61i)25-s − 27-s − 6.74i·29-s − 2.64·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.832 + 0.554i)5-s + 0.282i·7-s + 0.333·9-s − 1.61i·11-s − 0.811·13-s + (0.480 − 0.320i)15-s − 0.517i·17-s + 0.397i·19-s − 0.162i·21-s − 1.56i·23-s + (0.385 − 0.922i)25-s − 0.192·27-s − 1.25i·29-s − 0.475·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $-0.367 + 0.930i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ -0.367 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285830 - 0.420215i\)
\(L(\frac12)\) \(\approx\) \(0.285830 - 0.420215i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (1.86 - 1.23i)T \)
good7 \( 1 - 0.746iT - 7T^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 + 2.92T + 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + 7.49iT - 23T^{2} \)
29 \( 1 + 6.74iT - 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 7.44T + 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 - 7.72T + 53T^{2} \)
59 \( 1 - 6.85iT - 59T^{2} \)
61 \( 1 + 6.45iT - 61T^{2} \)
67 \( 1 - 7.44T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 - 5.85T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−10.77652312462492700535411103695, −10.10758387195692543215560675033, −8.765249820012737844576492568359, −8.010972658619322647172228189177, −6.94637047290979896608279860049, −6.10825253591034551670328422353, −5.02681864634932828086060195998, −3.81865239981444901718684137271, −2.66392667684989614103597525436, −0.33163425643839558471229567552, 1.66927737273400304016758006046, 3.61845864906280522997966773554, 4.69339049838246513439474849954, 5.34199320992128241234921129174, 7.05263979102276750366648852407, 7.31410471152193804536647912447, 8.558560799337582168554399272931, 9.624475417156758401082498438054, 10.34888161939627999755696808577, 11.43928719766667296606009289602

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