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Properties

Label 2-60-20.7-c3-0-7
Degree $2$
Conductor $60$
Sign $0.664 + 0.747i$
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  + (−2.57 − 1.16i)2-s + (−2.12 − 2.12i)3-s + (5.29 + 6.00i)4-s + (2.61 + 10.8i)5-s + (3 + 7.93i)6-s + (17.4 − 17.4i)7-s + (−6.65 − 21.6i)8-s + 8.99i·9-s + (5.91 − 31.0i)10-s − 24.6i·11-s + (1.50 − 23.9i)12-s + (51.2 − 51.2i)13-s + (−65.3 + 24.7i)14-s + (17.5 − 28.6i)15-s + (−8 + 63.4i)16-s + (48.7 + 48.7i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)2-s + (−0.408 − 0.408i)3-s + (0.661 + 0.750i)4-s + (0.233 + 0.972i)5-s + (0.204 + 0.540i)6-s + (0.943 − 0.943i)7-s + (−0.294 − 0.955i)8-s + 0.333i·9-s + (0.187 − 0.982i)10-s − 0.676i·11-s + (0.0361 − 0.576i)12-s + (1.09 − 1.09i)13-s + (−1.24 + 0.471i)14-s + (0.301 − 0.492i)15-s + (−0.125 + 0.992i)16-s + (0.694 + 0.694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.664 + 0.747i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.883288 - 0.396593i\)
\(L(\frac12)\) \(\approx\) \(0.883288 - 0.396593i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.57 + 1.16i)T \)
3 \( 1 + (2.12 + 2.12i)T \)
5 \( 1 + (-2.61 - 10.8i)T \)
good7 \( 1 + (-17.4 + 17.4i)T - 343iT^{2} \)
11 \( 1 + 24.6iT - 1.33e3T^{2} \)
13 \( 1 + (-51.2 + 51.2i)T - 2.19e3iT^{2} \)
17 \( 1 + (-48.7 - 48.7i)T + 4.91e3iT^{2} \)
19 \( 1 - 100.T + 6.85e3T^{2} \)
23 \( 1 + (52.5 + 52.5i)T + 1.21e4iT^{2} \)
29 \( 1 + 52.9iT - 2.43e4T^{2} \)
31 \( 1 - 180. iT - 2.97e4T^{2} \)
37 \( 1 + (-43.4 - 43.4i)T + 5.06e4iT^{2} \)
41 \( 1 - 250.T + 6.89e4T^{2} \)
43 \( 1 + (306. + 306. i)T + 7.95e4iT^{2} \)
47 \( 1 + (267. - 267. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-201. + 201. i)T - 1.48e5iT^{2} \)
59 \( 1 + 74.8T + 2.05e5T^{2} \)
61 \( 1 + 595.T + 2.26e5T^{2} \)
67 \( 1 + (613. - 613. i)T - 3.00e5iT^{2} \)
71 \( 1 - 293. iT - 3.57e5T^{2} \)
73 \( 1 + (34.7 - 34.7i)T - 3.89e5iT^{2} \)
79 \( 1 - 382.T + 4.93e5T^{2} \)
83 \( 1 + (457. + 457. i)T + 5.71e5iT^{2} \)
89 \( 1 - 8.16iT - 7.04e5T^{2} \)
97 \( 1 + (94.7 + 94.7i)T + 9.12e5iT^{2} \)
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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

−14.31613794370717138595677123528, −13.26770260276857603203519815830, −11.73620638145896285195923992193, −10.81895075941308705682524921912, −10.25074815462229833987516047223, −8.268856027630851683494896042958, −7.42346422711481934166702537095, −6.01174172255678775019615513696, −3.37501384714158898367084854462, −1.19587133171856896357498480491, 1.54950131117263406195993307333, 4.89709234424454608644488224338, 5.94237411448124017296950443047, 7.77916608342361899559784667914, 9.002542702610205806422190503790, 9.705386355864309506462512989932, 11.40340418988307774158371099416, 11.97787967712658172079498608273, 13.88084441745055152531159961412, 15.09758675639604655909126706821

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