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Properties

Label 2-60-15.14-c6-0-6
Degree $2$
Conductor $60$
Sign $0.933 - 0.359i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (24.4 − 11.3i)3-s + (124. + 8.26i)5-s + 577. i·7-s + (471. − 556. i)9-s + 1.96e3i·11-s − 2.09e3i·13-s + (3.14e3 − 1.21e3i)15-s − 1.15e3·17-s + 5.61e3·19-s + (6.55e3 + 1.41e4i)21-s − 3.38e3·23-s + (1.54e4 + 2.06e3i)25-s + (5.22e3 − 1.89e4i)27-s − 5.72e3i·29-s + 3.69e4·31-s + ⋯
L(s)  = 1  + (0.907 − 0.420i)3-s + (0.997 + 0.0661i)5-s + 1.68i·7-s + (0.646 − 0.763i)9-s + 1.47i·11-s − 0.951i·13-s + (0.933 − 0.359i)15-s − 0.234·17-s + 0.818·19-s + (0.707 + 1.52i)21-s − 0.278·23-s + (0.991 + 0.131i)25-s + (0.265 − 0.964i)27-s − 0.234i·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.933 - 0.359i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.933 - 0.359i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.84162 + 0.528673i\)
\(L(\frac12)\) \(\approx\) \(2.84162 + 0.528673i\)
\(L(4)\) 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 \( 1 + (-24.4 + 11.3i)T \)
5 \( 1 + (-124. - 8.26i)T \)
good7 \( 1 - 577. iT - 1.17e5T^{2} \)
11 \( 1 - 1.96e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.09e3iT - 4.82e6T^{2} \)
17 \( 1 + 1.15e3T + 2.41e7T^{2} \)
19 \( 1 - 5.61e3T + 4.70e7T^{2} \)
23 \( 1 + 3.38e3T + 1.48e8T^{2} \)
29 \( 1 + 5.72e3iT - 5.94e8T^{2} \)
31 \( 1 - 3.69e4T + 8.87e8T^{2} \)
37 \( 1 - 3.26e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.65e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.57e5T + 1.07e10T^{2} \)
53 \( 1 + 1.09e5T + 2.21e10T^{2} \)
59 \( 1 + 2.27e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.71e5T + 5.15e10T^{2} \)
67 \( 1 + 2.19e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.22e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.78e3iT - 1.51e11T^{2} \)
79 \( 1 - 7.94e5T + 2.43e11T^{2} \)
83 \( 1 + 2.13e5T + 3.26e11T^{2} \)
89 \( 1 - 1.24e6iT - 4.96e11T^{2} \)
97 \( 1 + 4.92e5iT - 8.32e11T^{2} \)
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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

−13.91411805381117853640041413470, −12.78194399599506225290648278017, −12.08438796038916510365948571799, −10.00105712733816044087204493164, −9.264852120164535301719938193296, −8.079647727161948952030198243768, −6.55453476951696308772850073885, −5.18519560885191925249818553533, −2.81261953029871668027981481553, −1.83709847140846487379757913636, 1.27299152717091183565495809030, 3.18359397416401529152971999468, 4.57969911277800491147822501209, 6.46987788010690202239591784004, 7.892904836721590487985322413908, 9.210015011880827766007610228792, 10.16624337768040715937162885783, 11.16033276718204935866485031611, 13.31139706119921619140994590565, 13.79571057463639109766750749600

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