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Properties

Label 2-60-20.19-c2-0-3
Degree $2$
Conductor $60$
Sign $0.0450 - 0.998i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.656 + 1.88i)2-s + 1.73·3-s + (−3.13 + 2.48i)4-s + (−3.27 + 3.77i)5-s + (1.13 + 3.27i)6-s + 9.55·7-s + (−6.74 − 4.29i)8-s + 2.99·9-s + (−9.28 − 3.70i)10-s − 9.92i·11-s + (−5.43 + 4.29i)12-s − 7.55i·13-s + (6.27 + 18.0i)14-s + (−5.67 + 6.54i)15-s + (3.68 − 15.5i)16-s + 17.1i·17-s + ⋯
L(s)  = 1  + (0.328 + 0.944i)2-s + 0.577·3-s + (−0.784 + 0.620i)4-s + (−0.654 + 0.755i)5-s + (0.189 + 0.545i)6-s + 1.36·7-s + (−0.843 − 0.537i)8-s + 0.333·9-s + (−0.928 − 0.370i)10-s − 0.902i·11-s + (−0.452 + 0.358i)12-s − 0.581i·13-s + (0.448 + 1.28i)14-s + (−0.378 + 0.436i)15-s + (0.230 − 0.973i)16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0450 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0450 - 0.998i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.0450 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07195 + 1.02473i\)
\(L(\frac12)\) \(\approx\) \(1.07195 + 1.02473i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.656 - 1.88i)T \)
3 \( 1 - 1.73T \)
5 \( 1 + (3.27 - 3.77i)T \)
good7 \( 1 - 9.55T + 49T^{2} \)
11 \( 1 + 9.92iT - 121T^{2} \)
13 \( 1 + 7.55iT - 169T^{2} \)
17 \( 1 - 17.1iT - 289T^{2} \)
19 \( 1 + 26.1iT - 361T^{2} \)
23 \( 1 + 1.67T + 529T^{2} \)
29 \( 1 + 0.350T + 841T^{2} \)
31 \( 1 - 46.0iT - 961T^{2} \)
37 \( 1 - 22.6iT - 1.36e3T^{2} \)
41 \( 1 + 77.2T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 - 14.0T + 2.20e3T^{2} \)
53 \( 1 - 22.6iT - 2.80e3T^{2} \)
59 \( 1 + 94.7iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 + 29.8T + 4.48e3T^{2} \)
71 \( 1 + 7.19iT - 5.04e3T^{2} \)
73 \( 1 - 34.3iT - 5.32e3T^{2} \)
79 \( 1 + 46.0iT - 6.24e3T^{2} \)
83 \( 1 - 24.1T + 6.88e3T^{2} \)
89 \( 1 - 100.T + 7.92e3T^{2} \)
97 \( 1 - 131. iT - 9.40e3T^{2} \)
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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

−15.03719153622697507577899352432, −14.30524887099817242193542668693, −13.32921610242223139309509579713, −11.80537173687935362383330555568, −10.60724053252316920222639707422, −8.596739074861972747520513027468, −8.009684354705356939162922559445, −6.72641944700508647733971289322, −4.94337561389568052467302108314, −3.37123330768847173465987819710, 1.78676721456718306322720195618, 4.05306091365102585056042690753, 5.06245046059594834524209294761, 7.70241667998069432171540694033, 8.771552468125978852607259120999, 9.967447050929617424449558024446, 11.51754396703250652407584911203, 12.11829963712505594469597310578, 13.40558127287845532450303828991, 14.46850310442382004915156459574

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