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Properties

Label 2-60-60.59-c3-0-12
Degree $2$
Conductor $60$
Sign $0.260 - 0.965i$
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 + (−8.56 + 11.9i)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 + (−16.7 − 38.0i)12-s + 63.3i·13-s + (−11.1 + 18.7i)14-s + (41.8 + 40.3i)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 + (−0.582 + 0.812i)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 + (−0.403 − 0.915i)12-s + 1.35i·13-s + (−0.211 + 0.358i)14-s + (0.719 + 0.694i)15-s + (−0.534 + 0.845i)16-s + 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.260 - 0.965i$
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: \((2,\ 60,\ (\ :3/2),\ 0.260 - 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.27117 + 0.973344i\)
\(L(\frac12)\) \(\approx\) \(1.27117 + 0.973344i\)
\(L(\frac{5}{2})\) 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

−14.79194274816007037470382360612, −14.00100697337839568859241254786, −13.22234358708191062335957198113, −10.92984488257695100218578200260, −9.842050475812860018102357410322, −8.889933724173766832360279838954, −7.64685963754372852017603692383, −6.51060595470853075658628760892, −4.72044870900598101873304653049, −2.23002291451460931812001514529, 1.57883565237047192068354671834, 3.18634118924841323631473630566, 5.14533979536069568245410759612, 7.74499432259691764992813526873, 8.487293213169890428385961945743, 9.772987600307644960346623984812, 10.52087368157517277777383100103, 12.35976319475800639897776168176, 13.12885609215424820784820814010, 13.97361142753846949778432060761

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