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Properties

Label 1-3724-3724.1283-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.451 - 0.892i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.456 − 0.889i)3-s + (−0.998 + 0.0498i)5-s + (−0.583 − 0.811i)9-s + (−0.0747 − 0.997i)11-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (−0.661 − 0.749i)17-s + (0.318 + 0.947i)23-s + (0.995 − 0.0995i)25-s + (−0.988 + 0.149i)27-s + (0.853 − 0.521i)29-s + 31-s + (−0.921 − 0.388i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯
L(s)  = 1  + (0.456 − 0.889i)3-s + (−0.998 + 0.0498i)5-s + (−0.583 − 0.811i)9-s + (−0.0747 − 0.997i)11-s + (−0.270 + 0.962i)13-s + (−0.411 + 0.911i)15-s + (−0.661 − 0.749i)17-s + (0.318 + 0.947i)23-s + (0.995 − 0.0995i)25-s + (−0.988 + 0.149i)27-s + (0.853 − 0.521i)29-s + 31-s + (−0.921 − 0.388i)33-s + (0.988 + 0.149i)37-s + (0.733 + 0.680i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.451 - 0.892i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.451 - 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7107137690 - 1.156059476i\)
\(L(\frac12)\) \(\approx\) \(0.7107137690 - 1.156059476i\)
\(L(1)\) \(\approx\) \(0.9134422918 - 0.3998126593i\)
\(L(1)\) \(\approx\) \(0.9134422918 - 0.3998126593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.456 - 0.889i)T \)
5 \( 1 + (-0.998 + 0.0498i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.270 + 0.962i)T \)
17 \( 1 + (-0.661 - 0.749i)T \)
23 \( 1 + (0.318 + 0.947i)T \)
29 \( 1 + (0.853 - 0.521i)T \)
31 \( 1 + T \)
37 \( 1 + (0.988 + 0.149i)T \)
41 \( 1 + (0.998 - 0.0498i)T \)
43 \( 1 + (0.797 - 0.603i)T \)
47 \( 1 + (-0.270 + 0.962i)T \)
53 \( 1 + (0.661 - 0.749i)T \)
59 \( 1 + (0.921 + 0.388i)T \)
61 \( 1 + (-0.853 + 0.521i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.0249 + 0.999i)T \)
73 \( 1 + (-0.969 - 0.246i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (0.411 - 0.911i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−19.08070455173803697839070503915, −18.10173563653509089207759554112, −17.41378014048420911266784178100, −16.683288825232234450277735919286, −15.85839123018849400163033542429, −15.48392674008158740597084770214, −14.781952121454255745147581391, −14.45194626213328167932452526147, −13.22871245495069223449304734468, −12.65660385070302749940602884631, −11.95081469077822349859413172679, −11.022373745483054610393657317711, −10.507585829404478365449930715687, −9.88654547586240470782171588708, −8.97426003943178337019329841393, −8.32817747950536687005118839937, −7.786643900450278250306701118368, −6.983728372936934284730958436031, −6.00827897583865240351306794981, −4.909442076965025278643995814567, −4.49393177734252251521305056589, −3.84444281679025808127237952883, −2.87290093742732561150130311057, −2.36079741699190457059397828895, −0.89958310459420610092783029529, 0.47821442242682586364405016324, 1.27043508703591380853375101894, 2.52085839560161632799828734329, 2.957514738634052861434511766383, 3.95702572783239584217446372347, 4.59341247537830097816718645813, 5.759560762034860209882965130168, 6.5158104545293774981797794708, 7.216277912234581411639745166365, 7.75576068851447520093706602653, 8.57326366262402533038333092840, 9.014032572593007840927372902391, 9.92282296687976856673063616988, 11.22901674629554324587010000423, 11.45059477638326481449358981982, 12.10160351005714167906802637999, 12.945158791911970994479010729797, 13.65475791368102852464458100186, 14.15671583977801009249551664147, 14.89332015500582134045056036883, 15.75567822693829279100800764007, 16.179338528503056537870633400579, 17.14415247800500293464051172860, 17.807248076661128830234188416181, 18.64804997639282340751628543527

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