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Properties

Label 1-7e2-49.5-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: $\chi_{49} (5, \cdot )$
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

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

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