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Properties

Label 2-60-12.11-c5-0-33
Degree $2$
Conductor $60$
Sign $0.693 + 0.720i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.57 − 0.966i)2-s + (14.2 − 6.40i)3-s + (30.1 − 10.7i)4-s − 25i·5-s + (73.0 − 49.4i)6-s + 68.6i·7-s + (157. − 89.1i)8-s + (160. − 182. i)9-s + (−24.1 − 139. i)10-s − 118.·11-s + (359. − 346. i)12-s − 360.·13-s + (66.3 + 382. i)14-s + (−160. − 355. i)15-s + (791. − 649. i)16-s + 1.87e3i·17-s + ⋯
L(s)  = 1  + (0.985 − 0.170i)2-s + (0.911 − 0.410i)3-s + (0.941 − 0.336i)4-s − 0.447i·5-s + (0.828 − 0.560i)6-s + 0.529i·7-s + (0.870 − 0.492i)8-s + (0.662 − 0.749i)9-s + (−0.0763 − 0.440i)10-s − 0.296·11-s + (0.720 − 0.693i)12-s − 0.590·13-s + (0.0904 + 0.521i)14-s + (−0.183 − 0.407i)15-s + (0.773 − 0.633i)16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.69696 - 1.57201i\)
\(L(\frac12)\) \(\approx\) \(3.69696 - 1.57201i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.57 + 0.966i)T \)
3 \( 1 + (-14.2 + 6.40i)T \)
5 \( 1 + 25iT \)
good7 \( 1 - 68.6iT - 1.68e4T^{2} \)
11 \( 1 + 118.T + 1.61e5T^{2} \)
13 \( 1 + 360.T + 3.71e5T^{2} \)
17 \( 1 - 1.87e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.31e3iT - 2.47e6T^{2} \)
23 \( 1 - 626.T + 6.43e6T^{2} \)
29 \( 1 - 5.24e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.58e3iT - 2.86e7T^{2} \)
37 \( 1 + 9.44e3T + 6.93e7T^{2} \)
41 \( 1 - 1.20e4iT - 1.15e8T^{2} \)
43 \( 1 - 3.92e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.47e4T + 2.29e8T^{2} \)
53 \( 1 - 2.25e4iT - 4.18e8T^{2} \)
59 \( 1 + 5.14e4T + 7.14e8T^{2} \)
61 \( 1 + 3.16e4T + 8.44e8T^{2} \)
67 \( 1 + 6.90e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.63e4T + 1.80e9T^{2} \)
73 \( 1 - 6.70e4T + 2.07e9T^{2} \)
79 \( 1 + 7.96e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.30e4T + 3.93e9T^{2} \)
89 \( 1 + 1.27e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.77e4T + 8.58e9T^{2} \)
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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

−13.85999431941789571590681997880, −12.80827498275132962351558547433, −12.25134270990820020961040226679, −10.64286385081988798125975020831, −9.147675622083190804315904777832, −7.82765814580852629673294156044, −6.42695394927662943118372262457, −4.82247012525584277996464946437, −3.19686732411261336857478923627, −1.73544327141757479194469248619, 2.41485903961277254360806331040, 3.72020817776355496881296898652, 5.09228485524881365341447671218, 6.99210092398861281288572256276, 7.917904310652941437416257586138, 9.727122762796010475665427765424, 10.82586925953994990463034322241, 12.18399307755653619329137362343, 13.58649319397025133229627409193, 14.09642390271856600646353591048

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