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Properties

Label 2-60-20.19-c4-0-19
Degree $2$
Conductor $60$
Sign $0.0157 + 0.999i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 + 3.87i)2-s − 5.19·3-s + (−14.0 + 7.66i)4-s + (11.6 − 22.1i)5-s + (−5.13 − 20.1i)6-s − 66.7·7-s + (−43.6 − 46.8i)8-s + 27·9-s + (97.2 + 23.1i)10-s − 100. i·11-s + (72.9 − 39.8i)12-s − 184. i·13-s + (−66.0 − 258. i)14-s + (−60.4 + 114. i)15-s + (138. − 215. i)16-s + 312. i·17-s + ⋯
L(s)  = 1  + (0.247 + 0.968i)2-s − 0.577·3-s + (−0.877 + 0.479i)4-s + (0.465 − 0.885i)5-s + (−0.142 − 0.559i)6-s − 1.36·7-s + (−0.681 − 0.732i)8-s + 0.333·9-s + (0.972 + 0.231i)10-s − 0.831i·11-s + (0.506 − 0.276i)12-s − 1.08i·13-s + (−0.336 − 1.31i)14-s + (−0.268 + 0.511i)15-s + (0.540 − 0.841i)16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0157 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0157 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0157 + 0.999i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.0157 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.337468 - 0.332209i\)
\(L(\frac12)\) \(\approx\) \(0.337468 - 0.332209i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 - 3.87i)T \)
3 \( 1 + 5.19T \)
5 \( 1 + (-11.6 + 22.1i)T \)
good7 \( 1 + 66.7T + 2.40e3T^{2} \)
11 \( 1 + 100. iT - 1.46e4T^{2} \)
13 \( 1 + 184. iT - 2.85e4T^{2} \)
17 \( 1 - 312. iT - 8.35e4T^{2} \)
19 \( 1 - 18.4iT - 1.30e5T^{2} \)
23 \( 1 + 983.T + 2.79e5T^{2} \)
29 \( 1 + 1.08e3T + 7.07e5T^{2} \)
31 \( 1 - 572. iT - 9.23e5T^{2} \)
37 \( 1 + 776. iT - 1.87e6T^{2} \)
41 \( 1 - 261.T + 2.82e6T^{2} \)
43 \( 1 - 808.T + 3.41e6T^{2} \)
47 \( 1 - 2.52e3T + 4.87e6T^{2} \)
53 \( 1 - 3.49e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.31e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.23e3T + 1.38e7T^{2} \)
67 \( 1 - 6.82e3T + 2.01e7T^{2} \)
71 \( 1 - 268. iT - 2.54e7T^{2} \)
73 \( 1 + 6.21e3iT - 2.83e7T^{2} \)
79 \( 1 + 9.28e3iT - 3.89e7T^{2} \)
83 \( 1 - 46.6T + 4.74e7T^{2} \)
89 \( 1 + 7.76e3T + 6.27e7T^{2} \)
97 \( 1 - 8.94e3iT - 8.85e7T^{2} \)
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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

−13.93132216019743949579095007666, −12.92102780372956319784796020084, −12.40435914583934009119301850384, −10.32274741375832793386923018070, −9.218455715419884124508830055004, −7.949656374099680617288557472922, −6.18242284704713397939507119203, −5.63631302195225652726137287103, −3.79213194452638860869922863267, −0.25408635099900855568223816911, 2.29750817697218976719553403686, 3.97360066755246241832560691860, 5.82032829687109851282258688093, 6.98945858484432785986507842454, 9.540491201153510469283462928136, 9.930795286399334565251537370684, 11.28449131876150820492442779587, 12.23280992356798817060935536202, 13.35936556428277514399130018185, 14.26199783197625153393786713467

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