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Properties

Label 2-60-60.59-c5-0-41
Degree $2$
Conductor $60$
Sign $-0.200 + 0.979i$
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  + (−3.28 + 4.60i)2-s + (3.33 − 15.2i)3-s + (−10.3 − 30.2i)4-s + (−17.3 + 53.1i)5-s + (59.1 + 65.4i)6-s + 121.·7-s + (173. + 51.9i)8-s + (−220. − 101. i)9-s + (−187. − 254. i)10-s − 442.·11-s + (−495. + 56.8i)12-s − 1.18e3i·13-s + (−398. + 557. i)14-s + (751. + 441. i)15-s + (−809. + 627. i)16-s − 1.03e3·17-s + ⋯
L(s)  = 1  + (−0.581 + 0.813i)2-s + (0.213 − 0.976i)3-s + (−0.323 − 0.946i)4-s + (−0.310 + 0.950i)5-s + (0.670 + 0.741i)6-s + 0.934·7-s + (0.958 + 0.286i)8-s + (−0.908 − 0.417i)9-s + (−0.592 − 0.805i)10-s − 1.10·11-s + (−0.993 + 0.114i)12-s − 1.94i·13-s + (−0.543 + 0.760i)14-s + (0.862 + 0.506i)15-s + (−0.790 + 0.612i)16-s − 0.866·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.200 + 0.979i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.200 + 0.979i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.449822 - 0.551159i\)
\(L(\frac12)\) \(\approx\) \(0.449822 - 0.551159i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.28 - 4.60i)T \)
3 \( 1 + (-3.33 + 15.2i)T \)
5 \( 1 + (17.3 - 53.1i)T \)
good7 \( 1 - 121.T + 1.68e4T^{2} \)
11 \( 1 + 442.T + 1.61e5T^{2} \)
13 \( 1 + 1.18e3iT - 3.71e5T^{2} \)
17 \( 1 + 1.03e3T + 1.41e6T^{2} \)
19 \( 1 + 1.29e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.19e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.78e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.53e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.28e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.10e3iT - 1.15e8T^{2} \)
43 \( 1 + 194.T + 1.47e8T^{2} \)
47 \( 1 + 5.08e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.52e4T + 4.18e8T^{2} \)
59 \( 1 - 3.39e4T + 7.14e8T^{2} \)
61 \( 1 - 2.29e3T + 8.44e8T^{2} \)
67 \( 1 - 4.39e4T + 1.35e9T^{2} \)
71 \( 1 - 9.41e3T + 1.80e9T^{2} \)
73 \( 1 + 1.27e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.33e4iT - 3.07e9T^{2} \)
83 \( 1 + 3.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.10e5iT - 5.58e9T^{2} \)
97 \( 1 + 5.72e4iT - 8.58e9T^{2} \)
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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

−13.97685393237803294442225871264, −12.94258649071270903192256469995, −11.20230691709951203535328532133, −10.37868419669391342030213791411, −8.379885731437917094290868882657, −7.80826644540303472062639198410, −6.69815610509653751740172520081, −5.27513800681190556109345891385, −2.52068377428126720647531415214, −0.38748000648307083670100873629, 1.92683480025102904013293698837, 4.02480082367461198900966952108, 4.96309035521510076219111703421, 7.82046306851376171227885622150, 8.772246134784153021178432454286, 9.653281326826010336414167287364, 11.07000067235181618563365644132, 11.69841942929511099815696222181, 13.15745047801096143395789170464, 14.31291400239246699101213373012

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