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Properties

Label 2-60-20.19-c4-0-7
Degree $2$
Conductor $60$
Sign $-0.105 - 0.994i$
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  + (−3.44 + 2.03i)2-s + 5.19·3-s + (7.71 − 14.0i)4-s + (−20.5 + 14.2i)5-s + (−17.8 + 10.5i)6-s + 51.0·7-s + (1.97 + 63.9i)8-s + 27·9-s + (41.5 − 90.9i)10-s + 37.8i·11-s + (40.0 − 72.8i)12-s + 201. i·13-s + (−175. + 103. i)14-s + (−106. + 74.2i)15-s + (−136. − 216. i)16-s + 370. i·17-s + ⋯
L(s)  = 1  + (−0.860 + 0.508i)2-s + 0.577·3-s + (0.482 − 0.876i)4-s + (−0.820 + 0.571i)5-s + (−0.497 + 0.293i)6-s + 1.04·7-s + (0.0308 + 0.999i)8-s + 0.333·9-s + (0.415 − 0.909i)10-s + 0.312i·11-s + (0.278 − 0.505i)12-s + 1.19i·13-s + (−0.896 + 0.530i)14-s + (−0.473 + 0.330i)15-s + (−0.535 − 0.844i)16-s + 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.105 - 0.994i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.760174 + 0.844912i\)
\(L(\frac12)\) \(\approx\) \(0.760174 + 0.844912i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.44 - 2.03i)T \)
3 \( 1 - 5.19T \)
5 \( 1 + (20.5 - 14.2i)T \)
good7 \( 1 - 51.0T + 2.40e3T^{2} \)
11 \( 1 - 37.8iT - 1.46e4T^{2} \)
13 \( 1 - 201. iT - 2.85e4T^{2} \)
17 \( 1 - 370. iT - 8.35e4T^{2} \)
19 \( 1 - 474. iT - 1.30e5T^{2} \)
23 \( 1 + 386.T + 2.79e5T^{2} \)
29 \( 1 - 1.16e3T + 7.07e5T^{2} \)
31 \( 1 + 1.45e3iT - 9.23e5T^{2} \)
37 \( 1 - 609. iT - 1.87e6T^{2} \)
41 \( 1 - 731.T + 2.82e6T^{2} \)
43 \( 1 - 1.82e3T + 3.41e6T^{2} \)
47 \( 1 + 1.39e3T + 4.87e6T^{2} \)
53 \( 1 + 3.05e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.23e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.50e3T + 1.38e7T^{2} \)
67 \( 1 - 5.33e3T + 2.01e7T^{2} \)
71 \( 1 + 5.69e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.46e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.14e3iT - 3.89e7T^{2} \)
83 \( 1 + 6.10e3T + 4.74e7T^{2} \)
89 \( 1 - 9.51e3T + 6.27e7T^{2} \)
97 \( 1 - 3.48e3iT - 8.85e7T^{2} \)
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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

−14.66848708934848309347520132329, −14.19156326431644579139899290395, −12.06411901578431489789376171769, −11.02361749166577803813540296461, −9.882344861224280800338998024372, −8.347921426751346610933146086224, −7.78872051628197442526724316386, −6.38565741471928481488379051919, −4.25672234774448294916059943196, −1.87036873322266905199408682433, 0.849082117444564902103574392927, 2.93409240207861775176791185206, 4.70176207804108693198780548882, 7.35886736315316319409233442432, 8.226311865604239649702848135303, 9.082657897795377806566837417538, 10.62201413870988591138534598820, 11.63285687045871354646555044699, 12.63224654142380629318466617417, 13.96768100467604087797918617688

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