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Properties

Label 2-60-5.3-c6-0-4
Degree $2$
Conductor $60$
Sign $-0.999 + 0.0422i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.0 − 11.0i)3-s + (−23.5 + 122. i)5-s + (312. − 312. i)7-s + 242. i·9-s − 524.·11-s + (−2.81e3 − 2.81e3i)13-s + (1.61e3 − 1.09e3i)15-s + (−6.55e3 + 6.55e3i)17-s + 7.55e3i·19-s − 6.89e3·21-s + (−8.76e3 − 8.76e3i)23-s + (−1.45e4 − 5.78e3i)25-s + (2.67e3 − 2.67e3i)27-s − 2.33e4i·29-s − 4.80e4·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.188 + 0.982i)5-s + (0.911 − 0.911i)7-s + 0.333i·9-s − 0.393·11-s + (−1.28 − 1.28i)13-s + (0.477 − 0.324i)15-s + (−1.33 + 1.33i)17-s + 1.10i·19-s − 0.744·21-s + (−0.720 − 0.720i)23-s + (−0.928 − 0.370i)25-s + (0.136 − 0.136i)27-s − 0.957i·29-s − 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0422i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.999 + 0.0422i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.999 + 0.0422i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00314720 - 0.148901i\)
\(L(\frac12)\) \(\approx\) \(0.00314720 - 0.148901i\)
\(L(4)\) 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 \( 1 + (11.0 + 11.0i)T \)
5 \( 1 + (23.5 - 122. i)T \)
good7 \( 1 + (-312. + 312. i)T - 1.17e5iT^{2} \)
11 \( 1 + 524.T + 1.77e6T^{2} \)
13 \( 1 + (2.81e3 + 2.81e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (6.55e3 - 6.55e3i)T - 2.41e7iT^{2} \)
19 \( 1 - 7.55e3iT - 4.70e7T^{2} \)
23 \( 1 + (8.76e3 + 8.76e3i)T + 1.48e8iT^{2} \)
29 \( 1 + 2.33e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.80e4T + 8.87e8T^{2} \)
37 \( 1 + (-3.41e4 + 3.41e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 3.08e4T + 4.75e9T^{2} \)
43 \( 1 + (2.17e4 + 2.17e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (-3.78e4 + 3.78e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-1.79e5 - 1.79e5i)T + 2.21e10iT^{2} \)
59 \( 1 - 2.00e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.96e4T + 5.15e10T^{2} \)
67 \( 1 + (-9.41e4 + 9.41e4i)T - 9.04e10iT^{2} \)
71 \( 1 + 2.81e5T + 1.28e11T^{2} \)
73 \( 1 + (4.54e5 + 4.54e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 1.34e5iT - 2.43e11T^{2} \)
83 \( 1 + (-5.35e5 - 5.35e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 7.82e4iT - 4.96e11T^{2} \)
97 \( 1 + (9.45e4 - 9.45e4i)T - 8.32e11iT^{2} \)
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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

−13.28633239963480065597271800982, −12.12228454536087311404167531297, −10.75132573794005753620739699331, −10.36957595522034810341141166162, −8.037506136417165360429675820050, −7.32246650735053879800201684274, −5.84808074124771141730602730060, −4.14844743998409728678122249808, −2.15738977929338814726991511695, −0.06028220828121684113195696981, 2.10038957036299460895206429008, 4.61787563016188618091646033712, 5.22419442454690075819623156153, 7.18016682658765084785464500529, 8.786295113897779511870477201636, 9.485637706685848015665787860602, 11.39833682551148242429109465953, 11.82753786289030492294281399766, 13.16706328845007821237472096553, 14.53936269693133638669372891231

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