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Properties

Label 2-60-60.23-c2-0-0
Degree $2$
Conductor $60$
Sign $-0.914 - 0.404i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.07 + 1.68i)2-s + (0.130 + 2.99i)3-s + (−1.67 − 3.63i)4-s + (−1.65 + 4.71i)5-s + (−5.18 − 3.01i)6-s + (−1.91 − 1.91i)7-s + (7.92 + 1.11i)8-s + (−8.96 + 0.782i)9-s + (−6.16 − 7.87i)10-s − 6.87·11-s + (10.6 − 5.47i)12-s + (12.2 + 12.2i)13-s + (5.29 − 1.15i)14-s + (−14.3 − 4.33i)15-s + (−10.4 + 12.1i)16-s + (9.47 + 9.47i)17-s + ⋯
L(s)  = 1  + (−0.539 + 0.841i)2-s + (0.0434 + 0.999i)3-s + (−0.417 − 0.908i)4-s + (−0.330 + 0.943i)5-s + (−0.864 − 0.502i)6-s + (−0.273 − 0.273i)7-s + (0.990 + 0.138i)8-s + (−0.996 + 0.0869i)9-s + (−0.616 − 0.787i)10-s − 0.624·11-s + (0.889 − 0.456i)12-s + (0.944 + 0.944i)13-s + (0.378 − 0.0826i)14-s + (−0.957 − 0.288i)15-s + (−0.651 + 0.758i)16-s + (0.557 + 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.158426 + 0.750858i\)
\(L(\frac12)\) \(\approx\) \(0.158426 + 0.750858i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 1.68i)T \)
3 \( 1 + (-0.130 - 2.99i)T \)
5 \( 1 + (1.65 - 4.71i)T \)
good7 \( 1 + (1.91 + 1.91i)T + 49iT^{2} \)
11 \( 1 + 6.87T + 121T^{2} \)
13 \( 1 + (-12.2 - 12.2i)T + 169iT^{2} \)
17 \( 1 + (-9.47 - 9.47i)T + 289iT^{2} \)
19 \( 1 - 33.2T + 361T^{2} \)
23 \( 1 + (-7.20 - 7.20i)T + 529iT^{2} \)
29 \( 1 - 2.29T + 841T^{2} \)
31 \( 1 - 12.1iT - 961T^{2} \)
37 \( 1 + (20.7 - 20.7i)T - 1.36e3iT^{2} \)
41 \( 1 + 50.9iT - 1.68e3T^{2} \)
43 \( 1 + (-15.1 + 15.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (26.7 - 26.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (-15.5 + 15.5i)T - 2.80e3iT^{2} \)
59 \( 1 + 63.0iT - 3.48e3T^{2} \)
61 \( 1 - 28.4T + 3.72e3T^{2} \)
67 \( 1 + (-32.4 - 32.4i)T + 4.48e3iT^{2} \)
71 \( 1 - 88.8T + 5.04e3T^{2} \)
73 \( 1 + (-71.1 - 71.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.1T + 6.24e3T^{2} \)
83 \( 1 + (58.6 + 58.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 41.1T + 7.92e3T^{2} \)
97 \( 1 + (-30.3 + 30.3i)T - 9.40e3iT^{2} \)
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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

−15.68065820539257170397565133139, −14.43259937985874799021672755370, −13.74660670886247878785565923730, −11.44487441658105864741228556407, −10.47597916438508777612391070004, −9.571400352363725144631290323831, −8.221331011949425758058362909025, −6.88747430716072578331374824809, −5.45617685278497226736838900520, −3.65655439224758989544589930694, 0.937553572646336409502750247042, 3.11246190813809614135330055371, 5.40990802668061878665931114208, 7.55898499256249721754521236668, 8.381747693857113861340966007424, 9.569037143089826650982252180288, 11.18973413165086065550761077531, 12.18029553414881794448547881166, 12.96388559994478832010699767955, 13.76665323452339455117810723035

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