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Properties

Label 2-60-60.47-c2-0-16
Degree $2$
Conductor $60$
Sign $0.898 + 0.438i$
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  + (1.75 − 0.961i)2-s + (2.86 + 0.903i)3-s + (2.15 − 3.37i)4-s + (−4.95 − 0.663i)5-s + (5.88 − 1.16i)6-s + (−7.30 + 7.30i)7-s + (0.535 − 7.98i)8-s + (7.36 + 5.17i)9-s + (−9.32 + 3.59i)10-s − 4.41·11-s + (9.20 − 7.69i)12-s + (7.53 − 7.53i)13-s + (−5.78 + 19.8i)14-s + (−13.5 − 6.37i)15-s + (−6.73 − 14.5i)16-s + (0.350 − 0.350i)17-s + ⋯
L(s)  = 1  + (0.876 − 0.480i)2-s + (0.953 + 0.301i)3-s + (0.538 − 0.842i)4-s + (−0.991 − 0.132i)5-s + (0.981 − 0.193i)6-s + (−1.04 + 1.04i)7-s + (0.0669 − 0.997i)8-s + (0.818 + 0.574i)9-s + (−0.932 + 0.359i)10-s − 0.401·11-s + (0.767 − 0.641i)12-s + (0.579 − 0.579i)13-s + (−0.413 + 1.41i)14-s + (−0.905 − 0.425i)15-s + (−0.420 − 0.907i)16-s + (0.0206 − 0.0206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 + 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.898 + 0.438i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.898 + 0.438i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.92585 - 0.444588i\)
\(L(\frac12)\) \(\approx\) \(1.92585 - 0.444588i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 + 0.961i)T \)
3 \( 1 + (-2.86 - 0.903i)T \)
5 \( 1 + (4.95 + 0.663i)T \)
good7 \( 1 + (7.30 - 7.30i)T - 49iT^{2} \)
11 \( 1 + 4.41T + 121T^{2} \)
13 \( 1 + (-7.53 + 7.53i)T - 169iT^{2} \)
17 \( 1 + (-0.350 + 0.350i)T - 289iT^{2} \)
19 \( 1 - 9.24T + 361T^{2} \)
23 \( 1 + (17.9 - 17.9i)T - 529iT^{2} \)
29 \( 1 + 5.52T + 841T^{2} \)
31 \( 1 + 48.1iT - 961T^{2} \)
37 \( 1 + (-3.39 - 3.39i)T + 1.36e3iT^{2} \)
41 \( 1 + 33.0iT - 1.68e3T^{2} \)
43 \( 1 + (-1.45 - 1.45i)T + 1.84e3iT^{2} \)
47 \( 1 + (-27.8 - 27.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (-52.6 - 52.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 46.1T + 3.72e3T^{2} \)
67 \( 1 + (32.1 - 32.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 + (72.2 - 72.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 55.9T + 6.24e3T^{2} \)
83 \( 1 + (-46.5 + 46.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 33.2T + 7.92e3T^{2} \)
97 \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \)
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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

−14.88443188904313326428799343919, −13.48319826920514110449906408052, −12.69731336270096458359810650847, −11.62252913349340621241211127517, −10.18912575208342347932769920965, −9.025243935115437429330996375037, −7.55381169227595769444278851926, −5.71618671815480250800836870355, −3.93697590137702052313772458507, −2.81348712807695141367387952100, 3.22176101477057418577412537003, 4.19412676366967503506498919087, 6.63380392603116426208058681146, 7.44628598212924949187711915962, 8.615697377963287977449594831137, 10.41088232237052271306955168969, 11.96632339463346402393752136339, 13.01604654052131441065759790093, 13.81753383668138706761807196991, 14.77671261668893155819792216266

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