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Properties

Label 1-740-740.47-r0-0-0
Degree $1$
Conductor $740$
Sign $0.339 + 0.940i$
Analytic cond. $3.43654$
Root an. cond. $3.43654$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s − 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − i·23-s − i·27-s − 29-s − 31-s + (0.866 + 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s − 11-s + (0.866 + 0.5i)13-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − i·23-s − i·27-s − 29-s − 31-s + (0.866 + 0.5i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.339 + 0.940i$
Analytic conductor: \(3.43654\)
Root analytic conductor: \(3.43654\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 740,\ (0:\ ),\ 0.339 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7459037642 + 0.5237031186i\)
\(L(\frac12)\) \(\approx\) \(0.7459037642 + 0.5237031186i\)
\(L(1)\) \(\approx\) \(0.8273529721 + 0.08329726345i\)
\(L(1)\) \(\approx\) \(0.8273529721 + 0.08329726345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 - T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.335129885926657135892985738594, −21.472592056797063698711103829544, −20.76055274676405034819013671613, −20.31114175618217494201308754968, −18.773709647677372100436592399759, −18.23078818259869530642320543421, −17.34801697225262640759550767452, −16.76551919783967131085054453678, −15.801943290396228423129313606328, −15.13990393276006633204453193809, −14.234700207353798804826489335087, −13.08418119169007199054498689764, −12.449683975321565119259173318066, −11.19076853453209915674633761509, −10.80951282597592369972219732484, −10.12289285225908264574689505963, −8.8768168978959276681005380176, −7.95564600565462521041618403498, −7.02368050494904989109398799494, −5.86374332789876007704386671598, −5.19362795091731012014049308019, −4.29366168719521681247044381473, −3.3403884971607796779568299387, −1.80788519274572333093428664590, −0.51275103133610689132904736194, 1.32887564459168370729566208941, 2.09716479624500992401870282667, 3.56684459556852822755316784018, 4.88138221422823514649412224443, 5.51896405605712612376999172572, 6.295666053799759068640613069991, 7.598963557138463151677490585797, 7.99584448277571387443680112684, 9.24378488620271841838231911602, 10.38476074272971842272507440071, 11.1841997005588756939413669782, 11.76680498289168964699340741926, 12.69665866912509040265265998828, 13.45558779211371948108246107724, 14.40591287160115506695216656681, 15.39594725775975142976029116013, 16.29205731552923601334001233609, 16.93051748974515463633385692800, 18.03652319785546932469836620161, 18.43414379587772527813784366529, 19.02607570648451997837527907501, 20.39590294517377800411907608730, 21.22368453378107454012127962052, 21.68999250129296700638831675786, 22.88077840849102870813811286774

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