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Properties

Label 2-60-60.59-c3-0-25
Degree $2$
Conductor $60$
Sign $-0.978 - 0.205i$
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  + (−2.76 − 0.583i)2-s + (−3.16 − 4.12i)3-s + (7.31 + 3.22i)4-s + (4.82 − 10.0i)5-s + (6.35 + 13.2i)6-s − 10.1·7-s + (−18.3 − 13.2i)8-s + (−6.95 + 26.0i)9-s + (−19.2 + 25.0i)10-s − 60.6·11-s + (−9.86 − 40.3i)12-s + 26.6i·13-s + (28.0 + 5.91i)14-s + (−56.8 + 12.0i)15-s + (43.1 + 47.2i)16-s − 32.4·17-s + ⋯
L(s)  = 1  + (−0.978 − 0.206i)2-s + (−0.609 − 0.793i)3-s + (0.914 + 0.403i)4-s + (0.431 − 0.901i)5-s + (0.432 + 0.901i)6-s − 0.547·7-s + (−0.812 − 0.583i)8-s + (−0.257 + 0.966i)9-s + (−0.608 + 0.793i)10-s − 1.66·11-s + (−0.237 − 0.971i)12-s + 0.567i·13-s + (0.536 + 0.112i)14-s + (−0.978 + 0.207i)15-s + (0.674 + 0.738i)16-s − 0.463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.978 - 0.205i$
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: \((2,\ 60,\ (\ :3/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0316107 + 0.304581i\)
\(L(\frac12)\) \(\approx\) \(0.0316107 + 0.304581i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.76 + 0.583i)T \)
3 \( 1 + (3.16 + 4.12i)T \)
5 \( 1 + (-4.82 + 10.0i)T \)
good7 \( 1 + 10.1T + 343T^{2} \)
11 \( 1 + 60.6T + 1.33e3T^{2} \)
13 \( 1 - 26.6iT - 2.19e3T^{2} \)
17 \( 1 + 32.4T + 4.91e3T^{2} \)
19 \( 1 + 106. iT - 6.85e3T^{2} \)
23 \( 1 - 88.0iT - 1.21e4T^{2} \)
29 \( 1 + 101. iT - 2.43e4T^{2} \)
31 \( 1 - 103. iT - 2.97e4T^{2} \)
37 \( 1 + 331. iT - 5.06e4T^{2} \)
41 \( 1 + 13.7iT - 6.89e4T^{2} \)
43 \( 1 + 72.5T + 7.95e4T^{2} \)
47 \( 1 + 463. iT - 1.03e5T^{2} \)
53 \( 1 + 282.T + 1.48e5T^{2} \)
59 \( 1 - 682.T + 2.05e5T^{2} \)
61 \( 1 - 140.T + 2.26e5T^{2} \)
67 \( 1 - 515.T + 3.00e5T^{2} \)
71 \( 1 + 38.7T + 3.57e5T^{2} \)
73 \( 1 + 747. iT - 3.89e5T^{2} \)
79 \( 1 + 862. iT - 4.93e5T^{2} \)
83 \( 1 - 534. iT - 5.71e5T^{2} \)
89 \( 1 - 507. iT - 7.04e5T^{2} \)
97 \( 1 - 376. iT - 9.12e5T^{2} \)
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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

−13.40316961417127159249154102855, −12.84608552971917446612253942917, −11.64973695890845273079440182898, −10.51320497211600864574924803387, −9.246475359177866968921851818914, −8.016024182646841853670921761720, −6.77922490862831608197626319255, −5.34109038165528953205522859946, −2.24071263917951385460586667607, −0.28954162897095074246905761232, 2.91616511941354611226167365266, 5.52126145009146020991360883067, 6.59046457897132103818857326756, 8.123237149586361244885711361726, 9.815025622894330155867802829530, 10.33528415084440635147205194303, 11.20547285557232996621163291790, 12.77001664998675789640359982679, 14.54269321337342521994097081204, 15.50346827918504222474584109896

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