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Properties

Label 2-60-20.3-c3-0-3
Degree $2$
Conductor $60$
Sign $-0.624 - 0.780i$
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  + (1.72 + 2.23i)2-s + (−2.12 + 2.12i)3-s + (−2.01 + 7.74i)4-s + (7.18 + 8.56i)5-s + (−8.41 − 1.07i)6-s + (−13.0 − 13.0i)7-s + (−20.8 + 8.87i)8-s − 8.99i·9-s + (−6.74 + 30.8i)10-s + 38.8i·11-s + (−12.1 − 20.7i)12-s + (41.6 + 41.6i)13-s + (6.62 − 51.7i)14-s + (−33.4 − 2.93i)15-s + (−55.8 − 31.2i)16-s + (80.8 − 80.8i)17-s + ⋯
L(s)  = 1  + (0.611 + 0.791i)2-s + (−0.408 + 0.408i)3-s + (−0.252 + 0.967i)4-s + (0.642 + 0.766i)5-s + (−0.572 − 0.0733i)6-s + (−0.703 − 0.703i)7-s + (−0.919 + 0.392i)8-s − 0.333i·9-s + (−0.213 + 0.976i)10-s + 1.06i·11-s + (−0.292 − 0.497i)12-s + (0.889 + 0.889i)13-s + (0.126 − 0.987i)14-s + (−0.575 − 0.0504i)15-s + (−0.872 − 0.487i)16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.624 - 0.780i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.716721 + 1.49139i\)
\(L(\frac12)\) \(\approx\) \(0.716721 + 1.49139i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.72 - 2.23i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-7.18 - 8.56i)T \)
good7 \( 1 + (13.0 + 13.0i)T + 343iT^{2} \)
11 \( 1 - 38.8iT - 1.33e3T^{2} \)
13 \( 1 + (-41.6 - 41.6i)T + 2.19e3iT^{2} \)
17 \( 1 + (-80.8 + 80.8i)T - 4.91e3iT^{2} \)
19 \( 1 - 23.9T + 6.85e3T^{2} \)
23 \( 1 + (-131. + 131. i)T - 1.21e4iT^{2} \)
29 \( 1 - 17.3iT - 2.43e4T^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-17.5 + 17.5i)T - 5.06e4iT^{2} \)
41 \( 1 + 228.T + 6.89e4T^{2} \)
43 \( 1 + (64.3 - 64.3i)T - 7.95e4iT^{2} \)
47 \( 1 + (49.7 + 49.7i)T + 1.03e5iT^{2} \)
53 \( 1 + (263. + 263. i)T + 1.48e5iT^{2} \)
59 \( 1 - 613.T + 2.05e5T^{2} \)
61 \( 1 + 406.T + 2.26e5T^{2} \)
67 \( 1 + (246. + 246. i)T + 3.00e5iT^{2} \)
71 \( 1 - 226. iT - 3.57e5T^{2} \)
73 \( 1 + (652. + 652. i)T + 3.89e5iT^{2} \)
79 \( 1 - 308.T + 4.93e5T^{2} \)
83 \( 1 + (-690. + 690. i)T - 5.71e5iT^{2} \)
89 \( 1 + 101. iT - 7.04e5T^{2} \)
97 \( 1 + (-921. + 921. i)T - 9.12e5iT^{2} \)
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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

−14.87225263846427943937369024534, −14.04208029331631386419960609273, −13.04217883438542727690759940790, −11.73245137734067385341630672585, −10.31213023787759730919022386809, −9.236554885918383328015587475481, −7.17621486381736103449749034522, −6.47613113780241647982725892556, −4.92788944161796981061018725444, −3.32769682743429130989285638491, 1.16026027054813663935218229730, 3.27179699047429251032819469650, 5.52794804144923259571706267941, 6.00323969717820439427388349363, 8.494332525521475423184943910997, 9.718898623804791961608401541666, 10.94439155272563393228123524443, 12.16708194664841151156018175838, 13.03896449740369656408444591247, 13.62041048216762302563805859498

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