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Properties

Label 2-60-60.59-c3-0-11
Degree $2$
Conductor $60$
Sign $0.811 - 0.584i$
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  + (−2.76 + 0.583i)2-s + (3.16 + 4.12i)3-s + (7.31 − 3.22i)4-s + (4.82 − 10.0i)5-s + (−11.1 − 9.55i)6-s + 10.1·7-s + (−18.3 + 13.2i)8-s + (−6.95 + 26.0i)9-s + (−7.47 + 30.7i)10-s + 60.6·11-s + (36.4 + 19.9i)12-s + 26.6i·13-s + (−28.0 + 5.91i)14-s + (56.8 − 12.0i)15-s + (43.1 − 47.2i)16-s − 32.4·17-s + ⋯
L(s)  = 1  + (−0.978 + 0.206i)2-s + (0.609 + 0.793i)3-s + (0.914 − 0.403i)4-s + (0.431 − 0.901i)5-s + (−0.759 − 0.650i)6-s + 0.547·7-s + (−0.812 + 0.583i)8-s + (−0.257 + 0.966i)9-s + (−0.236 + 0.971i)10-s + 1.66·11-s + (0.877 + 0.479i)12-s + 0.567i·13-s + (−0.536 + 0.112i)14-s + (0.978 − 0.207i)15-s + (0.674 − 0.738i)16-s − 0.463·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.811 - 0.584i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.22665 + 0.395728i\)
\(L(\frac12)\) \(\approx\) \(1.22665 + 0.395728i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.76 - 0.583i)T \)
3 \( 1 + (-3.16 - 4.12i)T \)
5 \( 1 + (-4.82 + 10.0i)T \)
good7 \( 1 - 10.1T + 343T^{2} \)
11 \( 1 - 60.6T + 1.33e3T^{2} \)
13 \( 1 - 26.6iT - 2.19e3T^{2} \)
17 \( 1 + 32.4T + 4.91e3T^{2} \)
19 \( 1 - 106. iT - 6.85e3T^{2} \)
23 \( 1 + 88.0iT - 1.21e4T^{2} \)
29 \( 1 + 101. iT - 2.43e4T^{2} \)
31 \( 1 + 103. iT - 2.97e4T^{2} \)
37 \( 1 + 331. iT - 5.06e4T^{2} \)
41 \( 1 + 13.7iT - 6.89e4T^{2} \)
43 \( 1 - 72.5T + 7.95e4T^{2} \)
47 \( 1 - 463. iT - 1.03e5T^{2} \)
53 \( 1 + 282.T + 1.48e5T^{2} \)
59 \( 1 + 682.T + 2.05e5T^{2} \)
61 \( 1 - 140.T + 2.26e5T^{2} \)
67 \( 1 + 515.T + 3.00e5T^{2} \)
71 \( 1 - 38.7T + 3.57e5T^{2} \)
73 \( 1 + 747. iT - 3.89e5T^{2} \)
79 \( 1 - 862. iT - 4.93e5T^{2} \)
83 \( 1 + 534. iT - 5.71e5T^{2} \)
89 \( 1 - 507. iT - 7.04e5T^{2} \)
97 \( 1 - 376. iT - 9.12e5T^{2} \)
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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

−14.71946700320996207179261207528, −14.07613494789139661843875660295, −12.18221805827197527173511871976, −10.98866659130222049694102290377, −9.600952332065687889394826745089, −9.011001642055122075572625539121, −7.964566225201492100653751554170, −6.09130856789088804915227130054, −4.31196058582629828617076669752, −1.76530837131268717616765969005, 1.55947914605572688447887433274, 3.16404596624403938425357274676, 6.43864550596450246852486688365, 7.23857320452691656274373253740, 8.625379650201564516900700962952, 9.586356332249854983379240330377, 11.07471077175354789712466071009, 11.93096418140114191135636043424, 13.42005152571255507591577921862, 14.55959085667298955667680181169

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