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Properties

Label 2-60-60.59-c5-0-29
Degree $2$
Conductor $60$
Sign $-0.885 + 0.464i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.20 − 2.22i)2-s + (−15.5 + 0.545i)3-s + (22.1 + 23.1i)4-s + (−17.3 + 53.1i)5-s + (82.2 + 31.7i)6-s + 106.·7-s + (−63.7 − 169. i)8-s + (242. − 16.9i)9-s + (208. − 238. i)10-s − 278.·11-s + (−357. − 348. i)12-s + 699. i·13-s + (−552. − 235. i)14-s + (240. − 837. i)15-s + (−44.7 + 1.02e3i)16-s − 1.73e3·17-s + ⋯
L(s)  = 1  + (−0.919 − 0.392i)2-s + (−0.999 + 0.0349i)3-s + (0.691 + 0.722i)4-s + (−0.309 + 0.950i)5-s + (0.932 + 0.360i)6-s + 0.819·7-s + (−0.352 − 0.935i)8-s + (0.997 − 0.0699i)9-s + (0.658 − 0.752i)10-s − 0.692·11-s + (−0.716 − 0.697i)12-s + 1.14i·13-s + (−0.753 − 0.321i)14-s + (0.276 − 0.961i)15-s + (−0.0436 + 0.999i)16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.885 + 0.464i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0205559 - 0.0833588i\)
\(L(\frac12)\) \(\approx\) \(0.0205559 - 0.0833588i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.20 + 2.22i)T \)
3 \( 1 + (15.5 - 0.545i)T \)
5 \( 1 + (17.3 - 53.1i)T \)
good7 \( 1 - 106.T + 1.68e4T^{2} \)
11 \( 1 + 278.T + 1.61e5T^{2} \)
13 \( 1 - 699. iT - 3.71e5T^{2} \)
17 \( 1 + 1.73e3T + 1.41e6T^{2} \)
19 \( 1 + 2.22e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.75e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.52e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.69e3iT - 2.86e7T^{2} \)
37 \( 1 - 977. iT - 6.93e7T^{2} \)
41 \( 1 + 1.74e4iT - 1.15e8T^{2} \)
43 \( 1 + 5.03e3T + 1.47e8T^{2} \)
47 \( 1 - 4.80e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.72e4T + 4.18e8T^{2} \)
59 \( 1 + 2.28e4T + 7.14e8T^{2} \)
61 \( 1 + 1.22e4T + 8.44e8T^{2} \)
67 \( 1 + 4.79e4T + 1.35e9T^{2} \)
71 \( 1 + 3.01e4T + 1.80e9T^{2} \)
73 \( 1 + 6.31e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.40e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.67e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.19e4iT - 5.58e9T^{2} \)
97 \( 1 - 2.38e4iT - 8.58e9T^{2} \)
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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

−13.38284428352063650417269298054, −11.87802070796753414172581752171, −11.13577057969053694534577805560, −10.54780688105621284023833742423, −8.999045952605520595636149754935, −7.41567224196215487130662076952, −6.54576175389371611533105086261, −4.42996566495210348937190796661, −2.20436468094544870059300719913, −0.06360614824055522619761376413, 1.44379136926987651960106180900, 4.83812093976844945484964194383, 5.86086208254448184387873060693, 7.57033829961711580855147672946, 8.452470968755405093520675453770, 10.01885518660300403461094755320, 11.02711549138277522261399374040, 12.01642891478042661626682474389, 13.24471601982847149718807519023, 15.11011959851062884366655624650

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