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Properties

Label 2-60-12.11-c5-0-4
Degree $2$
Conductor $60$
Sign $-0.946 - 0.321i$
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  + (4.53 + 3.38i)2-s + (−12.7 − 9.00i)3-s + (9.09 + 30.6i)4-s + 25i·5-s + (−27.2 − 83.8i)6-s − 20.4i·7-s + (−62.6 + 169. i)8-s + (80.8 + 229. i)9-s + (−84.6 + 113. i)10-s − 681.·11-s + (160. − 472. i)12-s − 767.·13-s + (69.3 − 92.8i)14-s + (225. − 318. i)15-s + (−858. + 558. i)16-s + 2.03e3i·17-s + ⋯
L(s)  = 1  + (0.801 + 0.598i)2-s + (−0.816 − 0.577i)3-s + (0.284 + 0.958i)4-s + 0.447i·5-s + (−0.308 − 0.951i)6-s − 0.158i·7-s + (−0.345 + 0.938i)8-s + (0.332 + 0.942i)9-s + (−0.267 + 0.358i)10-s − 1.69·11-s + (0.321 − 0.946i)12-s − 1.25·13-s + (0.0945 − 0.126i)14-s + (0.258 − 0.365i)15-s + (−0.838 + 0.544i)16-s + 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.946 - 0.321i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.946 - 0.321i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.167701 + 1.01469i\)
\(L(\frac12)\) \(\approx\) \(0.167701 + 1.01469i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.53 - 3.38i)T \)
3 \( 1 + (12.7 + 9.00i)T \)
5 \( 1 - 25iT \)
good7 \( 1 + 20.4iT - 1.68e4T^{2} \)
11 \( 1 + 681.T + 1.61e5T^{2} \)
13 \( 1 + 767.T + 3.71e5T^{2} \)
17 \( 1 - 2.03e3iT - 1.41e6T^{2} \)
19 \( 1 + 321. iT - 2.47e6T^{2} \)
23 \( 1 - 3.92e3T + 6.43e6T^{2} \)
29 \( 1 + 1.21e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.33e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.11e3T + 6.93e7T^{2} \)
41 \( 1 - 442. iT - 1.15e8T^{2} \)
43 \( 1 - 1.10e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.95e4T + 2.29e8T^{2} \)
53 \( 1 - 1.63e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.49e3T + 7.14e8T^{2} \)
61 \( 1 - 1.44e4T + 8.44e8T^{2} \)
67 \( 1 - 2.36e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.00e4T + 1.80e9T^{2} \)
73 \( 1 + 7.20e3T + 2.07e9T^{2} \)
79 \( 1 - 5.16e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.19e4T + 3.93e9T^{2} \)
89 \( 1 + 9.77e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.06e5T + 8.58e9T^{2} \)
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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

−14.66637390804390092661179006814, −13.13824246038709798446404325898, −12.77875924066512383239599095205, −11.37480086992837625466179875036, −10.37097342740274790459625371064, −8.012802204415875725607124101852, −7.16229399519715768971042257683, −5.87788449613761176376496399147, −4.74609328041817528331001219685, −2.57376829420837693176976560013, 0.39048583272164812842292473609, 2.81938427448507248814590330921, 4.86676863506735452863470689149, 5.30100569129354168867251152196, 7.14042317741820421865285292989, 9.389120917410850275627429694290, 10.33737670307040412070306776059, 11.42178918586535970948928199328, 12.40015993604663836732510696065, 13.28580855416374937404073205783

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