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Properties

Label 2-60-60.59-c3-0-1
Degree $2$
Conductor $60$
Sign $-0.888 + 0.459i$
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  + (1.43 + 2.43i)2-s + (−5.17 + 0.459i)3-s + (−3.85 + 7.00i)4-s + (−8.70 − 7.01i)5-s + (−8.56 − 11.9i)6-s − 7.71·7-s + (−22.6 + 0.684i)8-s + (26.5 − 4.75i)9-s + (4.57 − 31.2i)10-s − 38.0·11-s + (16.7 − 38.0i)12-s + 63.3i·13-s + (−11.1 − 18.7i)14-s + (48.2 + 32.3i)15-s + (−34.2 − 54.0i)16-s − 10.2·17-s + ⋯
L(s)  = 1  + (0.508 + 0.860i)2-s + (−0.996 + 0.0883i)3-s + (−0.482 + 0.875i)4-s + (−0.778 − 0.627i)5-s + (−0.582 − 0.812i)6-s − 0.416·7-s + (−0.999 + 0.0302i)8-s + (0.984 − 0.176i)9-s + (0.144 − 0.989i)10-s − 1.04·11-s + (0.403 − 0.915i)12-s + 1.35i·13-s + (−0.211 − 0.358i)14-s + (0.830 + 0.556i)15-s + (−0.534 − 0.845i)16-s − 0.146·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.888 + 0.459i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.888 + 0.459i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0847968 - 0.348697i\)
\(L(\frac12)\) \(\approx\) \(0.0847968 - 0.348697i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.43 - 2.43i)T \)
3 \( 1 + (5.17 - 0.459i)T \)
5 \( 1 + (8.70 + 7.01i)T \)
good7 \( 1 + 7.71T + 343T^{2} \)
11 \( 1 + 38.0T + 1.33e3T^{2} \)
13 \( 1 - 63.3iT - 2.19e3T^{2} \)
17 \( 1 + 10.2T + 4.91e3T^{2} \)
19 \( 1 - 99.5iT - 6.85e3T^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 - 197. iT - 2.43e4T^{2} \)
31 \( 1 - 13.9iT - 2.97e4T^{2} \)
37 \( 1 + 272. iT - 5.06e4T^{2} \)
41 \( 1 + 166. iT - 6.89e4T^{2} \)
43 \( 1 - 273.T + 7.95e4T^{2} \)
47 \( 1 - 69.1iT - 1.03e5T^{2} \)
53 \( 1 + 300.T + 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 + 439.T + 2.26e5T^{2} \)
67 \( 1 - 23.6T + 3.00e5T^{2} \)
71 \( 1 - 827.T + 3.57e5T^{2} \)
73 \( 1 - 152. iT - 3.89e5T^{2} \)
79 \( 1 - 421. iT - 4.93e5T^{2} \)
83 \( 1 - 709. iT - 5.71e5T^{2} \)
89 \( 1 - 1.43e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.08e3iT - 9.12e5T^{2} \)
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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

−15.59487581580563106577396070731, −14.21012775504713501287271926799, −12.69739291448232849727886934014, −12.30653914188564195152514386063, −10.90325370914102834477969141883, −9.192637228965946896163868859766, −7.76640009077201682250444972344, −6.54983648602729452605002536932, −5.17909009016430592303442515773, −4.04560432124773101943173458457, 0.23508384437504323915902113130, 3.05040134236427778365382409856, 4.78909802735935843937922187937, 6.12597348500100262346266772044, 7.70465955115950866906121093438, 9.867176940521897069497959839462, 10.80258137399568979615857994275, 11.55919017564345836033942669503, 12.72135432070187407142192702005, 13.48464988136589045935059409605

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