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Properties

Label 2-60-12.11-c3-0-5
Degree $2$
Conductor $60$
Sign $0.675 - 0.737i$
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  + (−0.622 − 2.75i)2-s + (1.95 + 4.81i)3-s + (−7.22 + 3.43i)4-s + 5i·5-s + (12.0 − 8.39i)6-s + 20.3i·7-s + (13.9 + 17.7i)8-s + (−19.3 + 18.8i)9-s + (13.7 − 3.11i)10-s + 15.2·11-s + (−30.6 − 28.0i)12-s + 27.7·13-s + (56.1 − 12.6i)14-s + (−24.0 + 9.77i)15-s + (40.4 − 49.6i)16-s − 92.5i·17-s + ⋯
L(s)  = 1  + (−0.220 − 0.975i)2-s + (0.376 + 0.926i)3-s + (−0.903 + 0.429i)4-s + 0.447i·5-s + (0.821 − 0.570i)6-s + 1.09i·7-s + (0.617 + 0.786i)8-s + (−0.716 + 0.697i)9-s + (0.436 − 0.0984i)10-s + 0.419·11-s + (−0.737 − 0.675i)12-s + 0.591·13-s + (1.07 − 0.241i)14-s + (−0.414 + 0.168i)15-s + (0.631 − 0.775i)16-s − 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.675 - 0.737i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.12273 + 0.494310i\)
\(L(\frac12)\) \(\approx\) \(1.12273 + 0.494310i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.622 + 2.75i)T \)
3 \( 1 + (-1.95 - 4.81i)T \)
5 \( 1 - 5iT \)
good7 \( 1 - 20.3iT - 343T^{2} \)
11 \( 1 - 15.2T + 1.33e3T^{2} \)
13 \( 1 - 27.7T + 2.19e3T^{2} \)
17 \( 1 + 92.5iT - 4.91e3T^{2} \)
19 \( 1 - 127. iT - 6.85e3T^{2} \)
23 \( 1 + 51.1T + 1.21e4T^{2} \)
29 \( 1 + 99.2iT - 2.43e4T^{2} \)
31 \( 1 - 25.8iT - 2.97e4T^{2} \)
37 \( 1 - 356.T + 5.06e4T^{2} \)
41 \( 1 - 292. iT - 6.89e4T^{2} \)
43 \( 1 + 521. iT - 7.95e4T^{2} \)
47 \( 1 - 573.T + 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 + 295.T + 2.05e5T^{2} \)
61 \( 1 - 326.T + 2.26e5T^{2} \)
67 \( 1 - 299. iT - 3.00e5T^{2} \)
71 \( 1 - 653.T + 3.57e5T^{2} \)
73 \( 1 - 504.T + 3.89e5T^{2} \)
79 \( 1 + 110. iT - 4.93e5T^{2} \)
83 \( 1 + 1.47e3T + 5.71e5T^{2} \)
89 \( 1 - 772. iT - 7.04e5T^{2} \)
97 \( 1 + 26.1T + 9.12e5T^{2} \)
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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

−14.58329859428845495794328000576, −13.72238934433933018948287190141, −12.12283858843863859785784230480, −11.28781952237072043229028375019, −10.04747338676496224215590401978, −9.187085163044426215380109336332, −8.093962261554006363002044351461, −5.60224512484618151484155635400, −3.92261826640751134536501715665, −2.51531616271297350742197864683, 0.989178836047957698201325676193, 4.12556435069751762470436448267, 6.08841423418318529856611035710, 7.18321534660008269593265667833, 8.252272927175410358769340562012, 9.296700315197890921339869754701, 10.91475929281161832473536430078, 12.71617702765279807403249413450, 13.49090762272257980975465727755, 14.33205445301058069548614395363

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