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Properties

Label 2-60-20.3-c3-0-2
Degree $2$
Conductor $60$
Sign $-0.0733 - 0.997i$
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  + (−1.79 + 2.18i)2-s + (2.12 − 2.12i)3-s + (−1.55 − 7.84i)4-s + (−3.87 + 10.4i)5-s + (0.829 + 8.44i)6-s + (17.0 + 17.0i)7-s + (19.9 + 10.6i)8-s − 8.99i·9-s + (−15.9 − 27.2i)10-s + 62.6i·11-s + (−19.9 − 13.3i)12-s + (−10.3 − 10.3i)13-s + (−67.8 + 6.66i)14-s + (14.0 + 30.4i)15-s + (−59.1 + 24.4i)16-s + (−15.8 + 15.8i)17-s + ⋯
L(s)  = 1  + (−0.634 + 0.772i)2-s + (0.408 − 0.408i)3-s + (−0.194 − 0.980i)4-s + (−0.346 + 0.937i)5-s + (0.0564 + 0.574i)6-s + (0.919 + 0.919i)7-s + (0.881 + 0.472i)8-s − 0.333i·9-s + (−0.504 − 0.863i)10-s + 1.71i·11-s + (−0.479 − 0.320i)12-s + (−0.221 − 0.221i)13-s + (−1.29 + 0.127i)14-s + (0.241 + 0.524i)15-s + (−0.924 + 0.381i)16-s + (−0.225 + 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0733 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0733 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0733 - 0.997i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.0733 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.766525 + 0.824955i\)
\(L(\frac12)\) \(\approx\) \(0.766525 + 0.824955i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.79 - 2.18i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
5 \( 1 + (3.87 - 10.4i)T \)
good7 \( 1 + (-17.0 - 17.0i)T + 343iT^{2} \)
11 \( 1 - 62.6iT - 1.33e3T^{2} \)
13 \( 1 + (10.3 + 10.3i)T + 2.19e3iT^{2} \)
17 \( 1 + (15.8 - 15.8i)T - 4.91e3iT^{2} \)
19 \( 1 - 40.5T + 6.85e3T^{2} \)
23 \( 1 + (-144. + 144. i)T - 1.21e4iT^{2} \)
29 \( 1 - 97.7iT - 2.43e4T^{2} \)
31 \( 1 + 3.67iT - 2.97e4T^{2} \)
37 \( 1 + (-158. + 158. i)T - 5.06e4iT^{2} \)
41 \( 1 + 330.T + 6.89e4T^{2} \)
43 \( 1 + (-59.6 + 59.6i)T - 7.95e4iT^{2} \)
47 \( 1 + (-0.929 - 0.929i)T + 1.03e5iT^{2} \)
53 \( 1 + (385. + 385. i)T + 1.48e5iT^{2} \)
59 \( 1 + 112.T + 2.05e5T^{2} \)
61 \( 1 - 420.T + 2.26e5T^{2} \)
67 \( 1 + (-505. - 505. i)T + 3.00e5iT^{2} \)
71 \( 1 + 713. iT - 3.57e5T^{2} \)
73 \( 1 + (-541. - 541. i)T + 3.89e5iT^{2} \)
79 \( 1 + 277.T + 4.93e5T^{2} \)
83 \( 1 + (-96.5 + 96.5i)T - 5.71e5iT^{2} \)
89 \( 1 + 370. iT - 7.04e5T^{2} \)
97 \( 1 + (-701. + 701. i)T - 9.12e5iT^{2} \)
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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

−14.84230753695492841879130770444, −14.44903327862505923093097752080, −12.66927832450682431948509290781, −11.32941908926335351102391183898, −10.04958441091702896399516206503, −8.737999568796447572941608915088, −7.60952352359965380930146549969, −6.71521118591653137296789794321, −4.93081715309904905937681407383, −2.17138282693750777898637884574, 1.04286440345743585627065944940, 3.49753942931928385574322499174, 4.85968498048048181257160110942, 7.63828332772621586296202364427, 8.500510056092448832609842043306, 9.522653306864591919246231061763, 11.01291423905260501354127402729, 11.59398644389560878963625217050, 13.26753901530662564071386839227, 13.91860436314325442268716782109

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