Properties

Label 2-60-5.2-c4-0-0
Degree $2$
Conductor $60$
Sign $-0.999 - 0.0329i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.67 + 3.67i)3-s + (4.94 + 24.5i)5-s + (−52.2 − 52.2i)7-s − 27i·9-s − 102.·11-s + (−203. + 203. i)13-s + (−108. − 71.8i)15-s + (−194. − 194. i)17-s + 178. i·19-s + 383.·21-s + (456. − 456. i)23-s + (−576. + 242. i)25-s + (99.2 + 99.2i)27-s + 1.06e3i·29-s + 552.·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.197 + 0.980i)5-s + (−1.06 − 1.06i)7-s − 0.333i·9-s − 0.849·11-s + (−1.20 + 1.20i)13-s + (−0.480 − 0.319i)15-s + (−0.673 − 0.673i)17-s + 0.494i·19-s + 0.870·21-s + (0.862 − 0.862i)23-s + (−0.921 + 0.387i)25-s + (0.136 + 0.136i)27-s + 1.27i·29-s + 0.574·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0329i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0329i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.999 - 0.0329i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00531807 + 0.323134i\)
\(L(\frac12)\) \(\approx\) \(0.00531807 + 0.323134i\)
\(L(3)\) 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 \( 1 + (3.67 - 3.67i)T \)
5 \( 1 + (-4.94 - 24.5i)T \)
good7 \( 1 + (52.2 + 52.2i)T + 2.40e3iT^{2} \)
11 \( 1 + 102.T + 1.46e4T^{2} \)
13 \( 1 + (203. - 203. i)T - 2.85e4iT^{2} \)
17 \( 1 + (194. + 194. i)T + 8.35e4iT^{2} \)
19 \( 1 - 178. iT - 1.30e5T^{2} \)
23 \( 1 + (-456. + 456. i)T - 2.79e5iT^{2} \)
29 \( 1 - 1.06e3iT - 7.07e5T^{2} \)
31 \( 1 - 552.T + 9.23e5T^{2} \)
37 \( 1 + (-223. - 223. i)T + 1.87e6iT^{2} \)
41 \( 1 - 3.06e3T + 2.82e6T^{2} \)
43 \( 1 + (2.30e3 - 2.30e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (964. + 964. i)T + 4.87e6iT^{2} \)
53 \( 1 + (1.61e3 - 1.61e3i)T - 7.89e6iT^{2} \)
59 \( 1 + 870. iT - 1.21e7T^{2} \)
61 \( 1 + 1.33e3T + 1.38e7T^{2} \)
67 \( 1 + (3.05e3 + 3.05e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.73e3T + 2.54e7T^{2} \)
73 \( 1 + (2.94e3 - 2.94e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 2.70e3iT - 3.89e7T^{2} \)
83 \( 1 + (-777. + 777. i)T - 4.74e7iT^{2} \)
89 \( 1 - 6.58e3iT - 6.27e7T^{2} \)
97 \( 1 + (4.13e3 + 4.13e3i)T + 8.85e7iT^{2} \)
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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

−14.82254672260832992765566674199, −13.86411125106712815210750990501, −12.69602961273402884072614590280, −11.21877667830428161826584686103, −10.29414710622388067328898689896, −9.476547730295539741063549861541, −7.27330934147482245502217390839, −6.50238938461038962088018973641, −4.59929043636936695657432808011, −2.91849018054907592396123581487, 0.17739548307531552967008497018, 2.56978428598701003118810173891, 5.06793063842064826217149550712, 6.04310913942516550112456851321, 7.75368076300301545670512655757, 9.078138715244960517114512941785, 10.17280578989574579681586979446, 11.82186766959521900497976450456, 12.88400519005733817799778398809, 13.14275208559492710697879734445

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