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Properties

Label 1-7e2-49.8-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: $\chi_{49} (8, \cdot )$
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

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

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