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Properties

Label 2-60-20.7-c3-0-12
Degree $2$
Conductor $60$
Sign $0.719 + 0.694i$
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  + (−0.431 + 2.79i)2-s + (−2.12 − 2.12i)3-s + (−7.62 − 2.40i)4-s + (2.54 − 10.8i)5-s + (6.84 − 5.01i)6-s + (9.77 − 9.77i)7-s + (10.0 − 20.2i)8-s + 8.99i·9-s + (29.3 + 11.8i)10-s − 32.5i·11-s + (11.0 + 21.2i)12-s + (−5.95 + 5.95i)13-s + (23.1 + 31.5i)14-s + (−28.4 + 17.6i)15-s + (52.3 + 36.7i)16-s + (−71.0 − 71.0i)17-s + ⋯
L(s)  = 1  + (−0.152 + 0.988i)2-s + (−0.408 − 0.408i)3-s + (−0.953 − 0.301i)4-s + (0.227 − 0.973i)5-s + (0.465 − 0.341i)6-s + (0.527 − 0.527i)7-s + (0.443 − 0.896i)8-s + 0.333i·9-s + (0.927 + 0.373i)10-s − 0.892i·11-s + (0.266 + 0.512i)12-s + (−0.127 + 0.127i)13-s + (0.441 + 0.602i)14-s + (−0.490 + 0.304i)15-s + (0.818 + 0.574i)16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.719 + 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.924052 - 0.373124i\)
\(L(\frac12)\) \(\approx\) \(0.924052 - 0.373124i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.431 - 2.79i)T \)
3 \( 1 + (2.12 + 2.12i)T \)
5 \( 1 + (-2.54 + 10.8i)T \)
good7 \( 1 + (-9.77 + 9.77i)T - 343iT^{2} \)
11 \( 1 + 32.5iT - 1.33e3T^{2} \)
13 \( 1 + (5.95 - 5.95i)T - 2.19e3iT^{2} \)
17 \( 1 + (71.0 + 71.0i)T + 4.91e3iT^{2} \)
19 \( 1 - 104.T + 6.85e3T^{2} \)
23 \( 1 + (83.3 + 83.3i)T + 1.21e4iT^{2} \)
29 \( 1 - 171. iT - 2.43e4T^{2} \)
31 \( 1 - 123. iT - 2.97e4T^{2} \)
37 \( 1 + (52.1 + 52.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 471.T + 6.89e4T^{2} \)
43 \( 1 + (-258. - 258. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-54.3 + 54.3i)T - 1.03e5iT^{2} \)
53 \( 1 + (-331. + 331. i)T - 1.48e5iT^{2} \)
59 \( 1 + 567.T + 2.05e5T^{2} \)
61 \( 1 - 832.T + 2.26e5T^{2} \)
67 \( 1 + (38.6 - 38.6i)T - 3.00e5iT^{2} \)
71 \( 1 + 534. iT - 3.57e5T^{2} \)
73 \( 1 + (418. - 418. i)T - 3.89e5iT^{2} \)
79 \( 1 + 76.6T + 4.93e5T^{2} \)
83 \( 1 + (-908. - 908. i)T + 5.71e5iT^{2} \)
89 \( 1 + 12.1iT - 7.04e5T^{2} \)
97 \( 1 + (613. + 613. i)T + 9.12e5iT^{2} \)
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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

−14.21361520466173197994263524126, −13.63671554837171439980400779883, −12.45082899260090832860676455665, −11.01536675912019190134269014819, −9.424031401898321219769218972974, −8.334694054238559392313257298222, −7.14800681340236402677532665848, −5.69147301867821130527812375634, −4.57440188629064102456212430708, −0.827528984274352141558768649615, 2.24054412345084697610691035387, 4.09418721646855787596403768008, 5.71892119266259059909082904352, 7.65767655984705980140958942266, 9.319840211810453804495953612801, 10.26113680174105624335451113317, 11.27405566932518474552590779767, 12.10168167970397791500390075557, 13.50044672098689336749124095113, 14.67479773703143384070041062725

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