Properties

Label 1-7e2-49.10-r1-0-0
Degree $1$
Conductor $49$
Sign $0.127 + 0.991i$
Analytic cond. $5.26578$
Root an. cond. $5.26578$
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  + (0.955 + 0.294i)2-s + (−0.365 + 0.930i)3-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)10-s + (−0.733 + 0.680i)11-s + (−0.826 + 0.563i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.365 + 0.930i)3-s + (0.826 + 0.563i)4-s + (0.988 − 0.149i)5-s + (−0.623 + 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.733 − 0.680i)9-s + (0.988 + 0.149i)10-s + (−0.733 + 0.680i)11-s + (−0.826 + 0.563i)12-s + (0.222 + 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.365 + 0.930i)16-s + (−0.0747 − 0.997i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.127 + 0.991i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ 0.127 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980309367 + 1.741369647i\)
\(L(\frac12)\) \(\approx\) \(1.980309367 + 1.741369647i\)
\(L(1)\) \(\approx\) \(1.640167836 + 0.8744918603i\)
\(L(1)\) \(\approx\) \(1.640167836 + 0.8744918603i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (0.222 + 0.974i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 - T \)
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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

−33.28845169841121410049209746930, −32.03956025887394588012058998685, −30.82168417458201829208129835939, −29.74293565598553338167185640362, −29.209194352435724719033431616995, −28.081247184613901743978507737673, −25.85144762429928744721162907988, −24.86094776581575977743361600613, −23.90319571219236525702665568989, −22.75979055442256861941851478425, −21.72736319031484652710899557318, −20.51433373615899241916197785895, −19.07655562791441200008521649790, −17.94244575433006717936864029248, −16.527727777068014570522300542419, −14.799723030864882527589528312377, −13.43050697107011991125962674170, −12.94198268788831187226915897005, −11.35607669336721180664248663498, −10.205355620711280578618418970120, −7.83087957176246023221459335467, −6.141377846261477098287292752698, −5.46836131814674616363246838761, −3.00141137354000752566608955154, −1.463860524682887391483790212177, 2.57259839427929327122150590891, 4.49078280374638788323632208988, 5.461210017907604256951570857857, 6.874432471852804902115788092599, 9.106945579190300336335469476600, 10.51195916784237431613573626576, 11.862483984664893690707140940605, 13.36658182512427170179892173109, 14.48897104355042445068350037258, 15.780231138580311941524990853114, 16.74194587016671722752850715196, 17.95936325995567428733180477172, 20.39425072684950642388159204771, 21.10093938217488561981130693084, 22.09537085052273705308355687210, 23.076268547275337510712959376366, 24.39879954769730947473148041433, 25.73435045112769348745259696985, 26.52109545067028947278921827455, 28.4946051351910236058230874212, 29.04161276149582137141958530333, 30.59197288045437445055222110185, 31.81604748001623351326382541293, 32.76454676194812386459128263979, 33.65952706490288314267229289257

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