Properties

Label 2-60-60.23-c2-0-13
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: $\chi_{60} (23, \cdot )$
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

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

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