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Properties

Label 2-60-12.11-c5-0-31
Degree $2$
Conductor $60$
Sign $0.451 + 0.892i$
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  + (4.48 − 3.45i)2-s + (10.3 + 11.6i)3-s + (8.18 − 30.9i)4-s − 25i·5-s + (86.6 + 16.4i)6-s − 180. i·7-s + (−70.0 − 166. i)8-s + (−28.0 + 241. i)9-s + (−86.2 − 112. i)10-s + 274.·11-s + (445. − 225. i)12-s + 983.·13-s + (−622. − 808. i)14-s + (291. − 259. i)15-s + (−890. − 506. i)16-s − 225. i·17-s + ⋯
L(s)  = 1  + (0.792 − 0.610i)2-s + (0.664 + 0.746i)3-s + (0.255 − 0.966i)4-s − 0.447i·5-s + (0.982 + 0.186i)6-s − 1.39i·7-s + (−0.387 − 0.922i)8-s + (−0.115 + 0.993i)9-s + (−0.272 − 0.354i)10-s + 0.682·11-s + (0.892 − 0.451i)12-s + 1.61·13-s + (−0.848 − 1.10i)14-s + (0.334 − 0.297i)15-s + (−0.869 − 0.494i)16-s − 0.189i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.451 + 0.892i$
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: \((2,\ 60,\ (\ :5/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.78592 - 1.71180i\)
\(L(\frac12)\) \(\approx\) \(2.78592 - 1.71180i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.48 + 3.45i)T \)
3 \( 1 + (-10.3 - 11.6i)T \)
5 \( 1 + 25iT \)
good7 \( 1 + 180. iT - 1.68e4T^{2} \)
11 \( 1 - 274.T + 1.61e5T^{2} \)
13 \( 1 - 983.T + 3.71e5T^{2} \)
17 \( 1 + 225. iT - 1.41e6T^{2} \)
19 \( 1 - 2.11e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.88e3T + 6.43e6T^{2} \)
29 \( 1 - 2.81e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.99e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.76e3T + 6.93e7T^{2} \)
41 \( 1 - 1.47e4iT - 1.15e8T^{2} \)
43 \( 1 - 8.63e3iT - 1.47e8T^{2} \)
47 \( 1 + 5.28e3T + 2.29e8T^{2} \)
53 \( 1 - 1.41e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.29e4T + 7.14e8T^{2} \)
61 \( 1 - 5.06e4T + 8.44e8T^{2} \)
67 \( 1 + 642. iT - 1.35e9T^{2} \)
71 \( 1 + 3.85e4T + 1.80e9T^{2} \)
73 \( 1 - 4.20e4T + 2.07e9T^{2} \)
79 \( 1 + 1.90e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.80e4T + 3.93e9T^{2} \)
89 \( 1 + 6.35e4iT - 5.58e9T^{2} \)
97 \( 1 + 4.78e4T + 8.58e9T^{2} \)
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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

−13.87881267365077578073749409372, −13.09785813513669738693872976207, −11.50359163354115410898834903307, −10.47728135754395941376852750596, −9.558134210262509052813725451971, −8.024336745043081822128574968670, −6.09691661965256421868800743630, −4.28688940208206472961818341611, −3.64528753148322529527348932385, −1.38036947422632193472732382608, 2.26774756679605107208617018348, 3.67999667943094816019029117224, 5.86894765186364658334540692009, 6.71261278556929986157746267353, 8.248637474061339286108012427652, 9.029514868469677031788221586967, 11.41296096131513108600362624217, 12.26289966029421984398869463324, 13.39652573635171304926131843442, 14.21943980502508946600632522342

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