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Properties

Label 2-60-60.59-c3-0-13
Degree $2$
Conductor $60$
Sign $-0.0867 - 0.996i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
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  + (1.43 + 2.43i)2-s + (5.17 + 0.459i)3-s + (−3.85 + 7.00i)4-s + (−8.70 + 7.01i)5-s + (6.32 + 13.2i)6-s + 7.71·7-s + (−22.6 + 0.684i)8-s + (26.5 + 4.75i)9-s + (−29.6 − 11.0i)10-s + 38.0·11-s + (−23.1 + 34.4i)12-s − 63.3i·13-s + (11.1 + 18.7i)14-s + (−48.2 + 32.3i)15-s + (−34.2 − 54.0i)16-s − 10.2·17-s + ⋯
L(s)  = 1  + (0.508 + 0.860i)2-s + (0.996 + 0.0883i)3-s + (−0.482 + 0.875i)4-s + (−0.778 + 0.627i)5-s + (0.430 + 0.902i)6-s + 0.416·7-s + (−0.999 + 0.0302i)8-s + (0.984 + 0.176i)9-s + (−0.936 − 0.350i)10-s + 1.04·11-s + (−0.557 + 0.829i)12-s − 1.35i·13-s + (0.211 + 0.358i)14-s + (−0.830 + 0.556i)15-s + (−0.534 − 0.845i)16-s − 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0867 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0867 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0867 - 0.996i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.0867 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.44984 + 1.58156i\)
\(L(\frac12)\) \(\approx\) \(1.44984 + 1.58156i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.43 - 2.43i)T \)
3 \( 1 + (-5.17 - 0.459i)T \)
5 \( 1 + (8.70 - 7.01i)T \)
good7 \( 1 - 7.71T + 343T^{2} \)
11 \( 1 - 38.0T + 1.33e3T^{2} \)
13 \( 1 + 63.3iT - 2.19e3T^{2} \)
17 \( 1 + 10.2T + 4.91e3T^{2} \)
19 \( 1 - 99.5iT - 6.85e3T^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 + 197. iT - 2.43e4T^{2} \)
31 \( 1 - 13.9iT - 2.97e4T^{2} \)
37 \( 1 - 272. iT - 5.06e4T^{2} \)
41 \( 1 - 166. iT - 6.89e4T^{2} \)
43 \( 1 + 273.T + 7.95e4T^{2} \)
47 \( 1 - 69.1iT - 1.03e5T^{2} \)
53 \( 1 + 300.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
61 \( 1 + 439.T + 2.26e5T^{2} \)
67 \( 1 + 23.6T + 3.00e5T^{2} \)
71 \( 1 + 827.T + 3.57e5T^{2} \)
73 \( 1 + 152. iT - 3.89e5T^{2} \)
79 \( 1 - 421. iT - 4.93e5T^{2} \)
83 \( 1 - 709. iT - 5.71e5T^{2} \)
89 \( 1 + 1.43e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.08e3iT - 9.12e5T^{2} \)
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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

−14.81352489649298498530955770869, −14.22492849579717082102814058670, −12.92613200603576803615663146517, −11.78218597877962781785975836567, −10.08360027473381779142482218679, −8.426111425723259578066150728203, −7.80646398908499777391486504335, −6.45103297466473964286113320320, −4.39059525601661110827676655019, −3.18480818316974661144971948821, 1.59317436900991581330051552794, 3.64373878541747859419306630885, 4.70326113411420348772653976309, 7.03945396530230334948168047843, 8.790631580980733744424966128373, 9.356328647392835804823336703193, 11.20874774010723949749325645907, 12.03663905860963546293931452579, 13.20019219914484003558657586023, 14.17329969236375731411241081779

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