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Properties

Label 2-60-60.59-c11-0-5
Degree $2$
Conductor $60$
Sign $0.935 - 0.354i$
Analytic cond. $46.1005$
Root an. cond. $6.78974$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (25.7 − 37.2i)2-s − 420. i·3-s + (−725. − 1.91e3i)4-s − 6.98e3·5-s + (−1.56e4 − 1.08e4i)6-s + (−8.99e4 − 2.22e4i)8-s − 1.77e5·9-s + (−1.79e5 + 2.60e5i)10-s + (−8.06e5 + 3.05e5i)12-s + 2.94e6i·15-s + (−3.14e6 + 2.77e6i)16-s − 2.58e5·17-s + (−4.55e6 + 6.59e6i)18-s − 1.09e7i·19-s + (5.06e6 + 1.33e7i)20-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s − 0.999i·3-s + (−0.354 − 0.935i)4-s − 0.999·5-s + (−0.822 − 0.568i)6-s + (−0.970 − 0.239i)8-s − 9-s + (−0.568 + 0.822i)10-s + (−0.935 + 0.354i)12-s + 0.999i·15-s + (−0.749 + 0.662i)16-s − 0.0440·17-s + (−0.568 + 0.822i)18-s − 1.01i·19-s + (0.354 + 0.935i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.935 - 0.354i$
Analytic conductor: \(46.1005\)
Root analytic conductor: \(6.78974\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :11/2),\ 0.935 - 0.354i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.199415 + 0.0365049i\)
\(L(\frac12)\) \(\approx\) \(0.199415 + 0.0365049i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-25.7 + 37.2i)T \)
3 \( 1 + 420. iT \)
5 \( 1 + 6.98e3T \)
good7 \( 1 + 1.97e9T^{2} \)
11 \( 1 + 2.85e11T^{2} \)
13 \( 1 - 1.79e12T^{2} \)
17 \( 1 + 2.58e5T + 3.42e13T^{2} \)
19 \( 1 + 1.09e7iT - 1.16e14T^{2} \)
23 \( 1 - 4.92e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.22e16T^{2} \)
31 \( 1 + 2.61e8iT - 2.54e16T^{2} \)
37 \( 1 - 1.77e17T^{2} \)
41 \( 1 - 5.50e17T^{2} \)
43 \( 1 + 9.29e17T^{2} \)
47 \( 1 + 1.15e8iT - 2.47e18T^{2} \)
53 \( 1 + 1.75e9T + 9.26e18T^{2} \)
59 \( 1 + 3.01e19T^{2} \)
61 \( 1 - 1.30e10T + 4.35e19T^{2} \)
67 \( 1 + 1.22e20T^{2} \)
71 \( 1 + 2.31e20T^{2} \)
73 \( 1 - 3.13e20T^{2} \)
79 \( 1 - 5.30e10iT - 7.47e20T^{2} \)
83 \( 1 - 6.18e10iT - 1.28e21T^{2} \)
89 \( 1 - 2.77e21T^{2} \)
97 \( 1 - 7.15e21T^{2} \)
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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

−12.77693672630878381800175350417, −11.65057083633409451788941613457, −11.16938302293806653186529597539, −9.395041812804373615663140118539, −8.026414685151222356576091204040, −6.76684124840093109796006179606, −5.30997676393656454076250778423, −3.79292438817615458629708035177, −2.53103449675216037866517554341, −1.07532282697412465242592220703, 0.05230714465806254412016473259, 3.07232712362621909049282462701, 4.10444347076214403787042579436, 5.07243583870785442941164992797, 6.53381126028247893241524907079, 8.005885622529706793083784046185, 8.844759430036204180346793324858, 10.43088186343815320574253445486, 11.70466905359933893872609117553, 12.65974673364193539857446221626

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