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Properties

Label 2-60-15.8-c3-0-0
Degree $2$
Conductor $60$
Sign $-0.0215 - 0.999i$
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  + (−5.05 − 1.18i)3-s + (8.06 + 7.74i)5-s + (−16.2 + 16.2i)7-s + (24.1 + 11.9i)9-s + 70.9i·11-s + (2.51 + 2.51i)13-s + (−31.6 − 48.7i)15-s + (−24.8 − 24.8i)17-s − 114. i·19-s + (101. − 62.9i)21-s + (−45.7 + 45.7i)23-s + (4.99 + 124. i)25-s + (−108. − 89.3i)27-s + 42.0·29-s + 193.·31-s + ⋯
L(s)  = 1  + (−0.973 − 0.228i)3-s + (0.721 + 0.692i)5-s + (−0.877 + 0.877i)7-s + (0.895 + 0.444i)9-s + 1.94i·11-s + (0.0535 + 0.0535i)13-s + (−0.543 − 0.839i)15-s + (−0.354 − 0.354i)17-s − 1.38i·19-s + (1.05 − 0.653i)21-s + (−0.414 + 0.414i)23-s + (0.0399 + 0.999i)25-s + (−0.770 − 0.637i)27-s + 0.269·29-s + 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0215 - 0.999i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.0215 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.640222 + 0.654171i\)
\(L(\frac12)\) \(\approx\) \(0.640222 + 0.654171i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.05 + 1.18i)T \)
5 \( 1 + (-8.06 - 7.74i)T \)
good7 \( 1 + (16.2 - 16.2i)T - 343iT^{2} \)
11 \( 1 - 70.9iT - 1.33e3T^{2} \)
13 \( 1 + (-2.51 - 2.51i)T + 2.19e3iT^{2} \)
17 \( 1 + (24.8 + 24.8i)T + 4.91e3iT^{2} \)
19 \( 1 + 114. iT - 6.85e3T^{2} \)
23 \( 1 + (45.7 - 45.7i)T - 1.21e4iT^{2} \)
29 \( 1 - 42.0T + 2.43e4T^{2} \)
31 \( 1 - 193.T + 2.97e4T^{2} \)
37 \( 1 + (37.4 - 37.4i)T - 5.06e4iT^{2} \)
41 \( 1 - 245. iT - 6.89e4T^{2} \)
43 \( 1 + (171. + 171. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-253. - 253. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-224. + 224. i)T - 1.48e5iT^{2} \)
59 \( 1 - 225.T + 2.05e5T^{2} \)
61 \( 1 - 183.T + 2.26e5T^{2} \)
67 \( 1 + (-316. + 316. i)T - 3.00e5iT^{2} \)
71 \( 1 + 225. iT - 3.57e5T^{2} \)
73 \( 1 + (-349. - 349. i)T + 3.89e5iT^{2} \)
79 \( 1 + 323. iT - 4.93e5T^{2} \)
83 \( 1 + (-553. + 553. i)T - 5.71e5iT^{2} \)
89 \( 1 + 351.T + 7.04e5T^{2} \)
97 \( 1 + (114. - 114. i)T - 9.12e5iT^{2} \)
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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

−15.14096818847045024701097390799, −13.49725853869766147477581616350, −12.57550840376437606844475283154, −11.58366650000925872391850943773, −10.17557282391366026911100884974, −9.435583621254353638232837436508, −7.12699063997611057400012631408, −6.34007852317555460368436742853, −4.91465835022530606225218502172, −2.33051560526464565715846217711, 0.73250120005283060200245619353, 3.87281720143494271188760636529, 5.65046837854643304843396715612, 6.47689179761193197054525519416, 8.466574377895034080024725579206, 9.936573299776463331867324653793, 10.68939650574534666632588662499, 12.08922261316440081251503506327, 13.20001865893654698091477792829, 13.97858976279123509218278271537

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