Properties

Label 2-60-60.23-c2-0-4
Degree $2$
Conductor $60$
Sign $0.842 - 0.538i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.81 − 0.837i)2-s + (2.69 + 1.32i)3-s + (2.59 + 3.04i)4-s + (−3.21 + 3.82i)5-s + (−3.78 − 4.65i)6-s + (3.54 + 3.54i)7-s + (−2.16 − 7.70i)8-s + (5.50 + 7.12i)9-s + (9.04 − 4.26i)10-s + 16.8·11-s + (2.96 + 11.6i)12-s + (−8.64 − 8.64i)13-s + (−3.46 − 9.40i)14-s + (−13.7 + 6.06i)15-s + (−2.51 + 15.8i)16-s + (−9.72 − 9.72i)17-s + ⋯
L(s)  = 1  + (−0.908 − 0.418i)2-s + (0.897 + 0.440i)3-s + (0.649 + 0.760i)4-s + (−0.642 + 0.765i)5-s + (−0.630 − 0.776i)6-s + (0.506 + 0.506i)7-s + (−0.270 − 0.962i)8-s + (0.611 + 0.791i)9-s + (0.904 − 0.426i)10-s + 1.53·11-s + (0.247 + 0.968i)12-s + (−0.665 − 0.665i)13-s + (−0.247 − 0.671i)14-s + (−0.914 + 0.404i)15-s + (−0.157 + 0.987i)16-s + (−0.572 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.842 - 0.538i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.975472 + 0.285242i\)
\(L(\frac12)\) \(\approx\) \(0.975472 + 0.285242i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.81 + 0.837i)T \)
3 \( 1 + (-2.69 - 1.32i)T \)
5 \( 1 + (3.21 - 3.82i)T \)
good7 \( 1 + (-3.54 - 3.54i)T + 49iT^{2} \)
11 \( 1 - 16.8T + 121T^{2} \)
13 \( 1 + (8.64 + 8.64i)T + 169iT^{2} \)
17 \( 1 + (9.72 + 9.72i)T + 289iT^{2} \)
19 \( 1 + 4.78T + 361T^{2} \)
23 \( 1 + (13.5 + 13.5i)T + 529iT^{2} \)
29 \( 1 - 14.8T + 841T^{2} \)
31 \( 1 + 14.0iT - 961T^{2} \)
37 \( 1 + (10.1 - 10.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (17.6 - 17.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (16.2 - 16.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (-53.9 - 53.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 36.6T + 5.04e3T^{2} \)
73 \( 1 + (12.6 + 12.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 88.4T + 6.24e3T^{2} \)
83 \( 1 + (-63.7 - 63.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + (85.3 - 85.3i)T - 9.40e3iT^{2} \)
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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

−15.05649911482889323632285345142, −14.15853135315977273701225451585, −12.33764389862005037451727424414, −11.35716633958091983776733332296, −10.21824987228935586670064395976, −9.060570934075005822135981443929, −8.078881972639456081000365213308, −6.89432184609754711658811500091, −4.03251489410526350204951969896, −2.50164257519799672690372644400, 1.49165401967167258213315030851, 4.25267361694592023034897630245, 6.63845763999343266851380844318, 7.74007914709630921245356926764, 8.735431569402024647299301737326, 9.583817789938855442823449204722, 11.34183584513957689235813283405, 12.37444695690101318305289059167, 14.02974705938797626223721329789, 14.74542899767629457677963739458

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