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Properties

Label 2-60-4.3-c2-0-3
Degree $2$
Conductor $60$
Sign $0.911 - 0.410i$
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  + (1.67 + 1.08i)2-s − 1.73i·3-s + (1.64 + 3.64i)4-s + 2.23·5-s + (1.87 − 2.90i)6-s − 0.596i·7-s + (−1.19 + 7.91i)8-s − 2.99·9-s + (3.75 + 2.42i)10-s − 9.27i·11-s + (6.31 − 2.84i)12-s − 23.5·13-s + (0.647 − 1.00i)14-s − 3.87i·15-s + (−10.5 + 11.9i)16-s + 3.97·17-s + ⋯
L(s)  = 1  + (0.839 + 0.542i)2-s − 0.577i·3-s + (0.410 + 0.911i)4-s + 0.447·5-s + (0.313 − 0.484i)6-s − 0.0852i·7-s + (−0.149 + 0.988i)8-s − 0.333·9-s + (0.375 + 0.242i)10-s − 0.843i·11-s + (0.526 − 0.237i)12-s − 1.80·13-s + (0.0462 − 0.0715i)14-s − 0.258i·15-s + (−0.662 + 0.749i)16-s + 0.233·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.911 - 0.410i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.911 - 0.410i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.75774 + 0.377883i\)
\(L(\frac12)\) \(\approx\) \(1.75774 + 0.377883i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.67 - 1.08i)T \)
3 \( 1 + 1.73iT \)
5 \( 1 - 2.23T \)
good7 \( 1 + 0.596iT - 49T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 + 23.5T + 169T^{2} \)
17 \( 1 - 3.97T + 289T^{2} \)
19 \( 1 - 7.04iT - 361T^{2} \)
23 \( 1 + 32.0iT - 529T^{2} \)
29 \( 1 - 35.6T + 841T^{2} \)
31 \( 1 - 59.2iT - 961T^{2} \)
37 \( 1 + 5.38T + 1.36e3T^{2} \)
41 \( 1 - 40.0T + 1.68e3T^{2} \)
43 \( 1 + 36.1iT - 1.84e3T^{2} \)
47 \( 1 - 74.0iT - 2.20e3T^{2} \)
53 \( 1 + 2.55T + 2.80e3T^{2} \)
59 \( 1 + 36.4iT - 3.48e3T^{2} \)
61 \( 1 + 8.73T + 3.72e3T^{2} \)
67 \( 1 + 69.7iT - 4.48e3T^{2} \)
71 \( 1 - 59.2iT - 5.04e3T^{2} \)
73 \( 1 + 83.0T + 5.32e3T^{2} \)
79 \( 1 + 65.8iT - 6.24e3T^{2} \)
83 \( 1 + 129. iT - 6.88e3T^{2} \)
89 \( 1 + 130.T + 7.92e3T^{2} \)
97 \( 1 - 93.1T + 9.40e3T^{2} \)
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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

−14.44540313398092451601713020766, −14.09345940105196737697623993639, −12.69761140035827819782207273276, −12.07222522013281140403499057071, −10.49860601051796777610229499492, −8.682525459835767074132984446258, −7.39392029482911805800171134180, −6.24660221299313911241017639546, −4.89052710190228291963232149969, −2.75098712081465627649227835374, 2.50579013517607124162787433933, 4.43052892972481700979321892961, 5.56116431538507416089103578025, 7.24603429283610330005194238906, 9.538857673506610035847107003068, 10.08096040694238366000585343752, 11.54011048322057777993934434244, 12.48371982152967120897995157645, 13.68024113085065703510549624421, 14.77884484210894326496296461998

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