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Properties

Label 2-62-31.2-c1-0-3
Degree $2$
Conductor $62$
Sign $0.978 - 0.207i$
Analytic cond. $0.495072$
Root an. cond. $0.703613$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (1.25 − 0.914i)3-s + (0.309 + 0.951i)4-s − 4.13·5-s + 1.55·6-s + (−0.5 − 1.53i)7-s + (−0.309 + 0.951i)8-s + (−0.179 + 0.551i)9-s + (−3.34 − 2.43i)10-s + (1.14 + 3.51i)11-s + (1.25 + 0.914i)12-s + (3.20 − 2.32i)13-s + (0.5 − 1.53i)14-s + (−5.20 + 3.78i)15-s + (−0.809 + 0.587i)16-s + (1.27 − 3.93i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.726 − 0.527i)3-s + (0.154 + 0.475i)4-s − 1.84·5-s + 0.635·6-s + (−0.188 − 0.581i)7-s + (−0.109 + 0.336i)8-s + (−0.0596 + 0.183i)9-s + (−1.05 − 0.768i)10-s + (0.343 + 1.05i)11-s + (0.363 + 0.263i)12-s + (0.888 − 0.645i)13-s + (0.133 − 0.411i)14-s + (−1.34 + 0.976i)15-s + (−0.202 + 0.146i)16-s + (0.309 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $0.978 - 0.207i$
Analytic conductor: \(0.495072\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{62} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 62,\ (\ :1/2),\ 0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11658 + 0.117005i\)
\(L(\frac12)\) \(\approx\) \(1.11658 + 0.117005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-2.85 - 4.78i)T \)
good3 \( 1 + (-1.25 + 0.914i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + 4.13T + 5T^{2} \)
7 \( 1 + (0.5 + 1.53i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.14 - 3.51i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.20 + 2.32i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.27 + 3.93i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.62 + 2.63i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.306 - 0.944i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.00 + 2.18i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 + (-1.84 - 1.34i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.399 + 0.290i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.5 - 1.08i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.0524 - 0.161i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.1 - 7.35i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 3.34T + 61T^{2} \)
67 \( 1 - 1.47T + 67T^{2} \)
71 \( 1 + (-2.55 + 7.87i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.547 + 1.68i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.87 - 5.77i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.35 - 4.61i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (5.81 + 17.9i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.291 + 0.898i)T + (-78.4 + 57.0i)T^{2} \)
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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

−15.12654262686297396307749234109, −14.00041563904659805155379749685, −12.94574045651679090736445001780, −11.99274317553965789807614707208, −10.81500744471092254649233692798, −8.682675528728950075349764145989, −7.66871672961611368885983473463, −7.00759551180591823631475234829, −4.59140907646697592628821373139, −3.31361775191376484645838891897, 3.42614523253195833193741103486, 4.06637469604218327145211726045, 6.25572019484939935429637141249, 8.208331968190805234130787465840, 8.943367631792106227957380668089, 10.75737000977214053159066424054, 11.69418117705498165616334937112, 12.56648497941368419481259669619, 14.11403293300034102366923957140, 15.03545611675810170439482063700

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