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Properties

Label 1-7e2-49.43-r0-0-0
Degree $1$
Conductor $49$
Sign $0.991 - 0.127i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
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  + (−0.222 − 0.974i)2-s + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.900 − 0.433i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + 18-s + 19-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s + (0.623 − 0.781i)6-s + (0.623 + 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)11-s + (−0.900 − 0.433i)12-s + (−0.222 − 0.974i)13-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + 18-s + 19-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ 0.991 - 0.127i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8925935616 - 0.05730640801i\)
\(L(\frac12)\) \(\approx\) \(0.8925935616 - 0.05730640801i\)
\(L(1)\) \(\approx\) \(1.023791681 - 0.1035073225i\)
\(L(1)\) \(\approx\) \(1.023791681 - 0.1035073225i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−33.7586402785314095226511377705, −32.79992692197996447256124499653, −31.650139144086965322686406188411, −30.79235449388928643905926871926, −29.0516744777387529061367890586, −28.1695671735905104999073136944, −26.38726665206773944692869631269, −25.71333855524619613217332334415, −24.449357741023211786280714893274, −24.04643015160435665677085356863, −22.44760611143481923303262050493, −20.72883789426959981598460265955, −19.50552687073716495469097634523, −18.120983076020760798995711106071, −17.3265759923928492554344157095, −15.88065114099585819791059205787, −14.44054181289359708980418807457, −13.48170480399353186470506898379, −12.370869314001565968481983923554, −9.72668240295155668674463812144, −8.79318709247798448521392914821, −7.46284207600816438205764074837, −6.202180298696978550086389021789, −4.52511060061827775425546967928, −1.77234304749994552732478436416, 2.46121909002405190843541144176, 3.53486273581514854921747657685, 5.39411848863421672343866881297, 7.903242996095945018416707250432, 9.33500199085899441776734207919, 10.32667693160530458365649614392, 11.32851541908202796292381943724, 13.3797584508276554534797991436, 14.15202604980070799391047558128, 15.71837203770491441544492656087, 17.41258133440542292568578254281, 18.59471216693562799002585529763, 19.80724972333966211464886398121, 20.870735425573394102993561076237, 21.948056618321603413857521129859, 22.58204270962316004451630676130, 24.824041634533836280752625376862, 26.293442135135634242206740315320, 26.7518696483887648358142765259, 28.01954143914513340556764756101, 29.344605885207215478742392866966, 30.265781645769383907373550148000, 31.44866870450242855807956723444, 32.40643545223126404738716346732, 33.63909587407967168707063733104

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