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Properties

Label 2-60-4.3-c4-0-5
Degree $2$
Conductor $60$
Sign $0.280 - 0.959i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95 − 0.566i)2-s + 5.19i·3-s + (15.3 + 4.48i)4-s + 11.1·5-s + (2.94 − 20.5i)6-s − 24.1i·7-s + (−58.2 − 26.4i)8-s − 27·9-s + (−44.2 − 6.32i)10-s + 227. i·11-s + (−23.2 + 79.8i)12-s + 285.·13-s + (−13.6 + 95.5i)14-s + 58.0i·15-s + (215. + 137. i)16-s − 301.·17-s + ⋯
L(s)  = 1  + (−0.989 − 0.141i)2-s + 0.577i·3-s + (0.959 + 0.280i)4-s + 0.447·5-s + (0.0816 − 0.571i)6-s − 0.492i·7-s + (−0.910 − 0.413i)8-s − 0.333·9-s + (−0.442 − 0.0632i)10-s + 1.87i·11-s + (−0.161 + 0.554i)12-s + 1.68·13-s + (−0.0697 + 0.487i)14-s + 0.258i·15-s + (0.843 + 0.537i)16-s − 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.280 - 0.959i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.829095 + 0.621710i\)
\(L(\frac12)\) \(\approx\) \(0.829095 + 0.621710i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.95 + 0.566i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 - 11.1T \)
good7 \( 1 + 24.1iT - 2.40e3T^{2} \)
11 \( 1 - 227. iT - 1.46e4T^{2} \)
13 \( 1 - 285.T + 2.85e4T^{2} \)
17 \( 1 + 301.T + 8.35e4T^{2} \)
19 \( 1 - 674. iT - 1.30e5T^{2} \)
23 \( 1 - 459. iT - 2.79e5T^{2} \)
29 \( 1 - 146.T + 7.07e5T^{2} \)
31 \( 1 + 702. iT - 9.23e5T^{2} \)
37 \( 1 + 100.T + 1.87e6T^{2} \)
41 \( 1 - 1.10e3T + 2.82e6T^{2} \)
43 \( 1 + 811. iT - 3.41e6T^{2} \)
47 \( 1 + 1.19e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.29e3T + 7.89e6T^{2} \)
59 \( 1 - 3.14e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.02e3T + 1.38e7T^{2} \)
67 \( 1 + 2.69e3iT - 2.01e7T^{2} \)
71 \( 1 + 8.36e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.72e3T + 2.83e7T^{2} \)
79 \( 1 + 2.83e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.62e3iT - 4.74e7T^{2} \)
89 \( 1 + 245.T + 6.27e7T^{2} \)
97 \( 1 - 1.57e3T + 8.85e7T^{2} \)
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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

−14.91341245193895349248914243537, −13.39365439634573423633396140523, −12.03262141182441961438377151010, −10.71786279279555594849829523356, −9.972583434686065461928446668281, −8.908452619694429689473305156929, −7.51126093004994828275304403851, −6.08993414726103451011872022944, −3.95014789875492532050504756552, −1.75536697496086105722650214915, 0.847909409679964641914456078194, 2.75009342502109956095300959502, 5.85426829536090411799026151766, 6.69055511394874963039286082866, 8.584330479631704824884337186043, 8.852987524583676432881241982340, 10.86329668972882701629276121208, 11.36435338844277923879784753178, 13.08444351665613506961065664664, 14.00390070765754850597152476005

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