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Properties

Label 1-35e2-1225.239-r0-0-0
Degree $1$
Conductor $1225$
Sign $-0.995 + 0.0921i$
Analytic cond. $5.68887$
Root an. cond. $5.68887$
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  + (0.995 + 0.0896i)2-s + (0.0448 + 0.998i)3-s + (0.983 + 0.178i)4-s + (−0.0448 + 0.998i)6-s + (0.963 + 0.266i)8-s + (−0.995 + 0.0896i)9-s + (−0.995 − 0.0896i)11-s + (−0.134 + 0.990i)12-s + (−0.858 + 0.512i)13-s + (0.936 + 0.351i)16-s + (−0.983 + 0.178i)17-s − 18-s + (−0.809 + 0.587i)19-s + (−0.983 − 0.178i)22-s + (−0.134 − 0.990i)23-s + (−0.222 + 0.974i)24-s + ⋯
L(s)  = 1  + (0.995 + 0.0896i)2-s + (0.0448 + 0.998i)3-s + (0.983 + 0.178i)4-s + (−0.0448 + 0.998i)6-s + (0.963 + 0.266i)8-s + (−0.995 + 0.0896i)9-s + (−0.995 − 0.0896i)11-s + (−0.134 + 0.990i)12-s + (−0.858 + 0.512i)13-s + (0.936 + 0.351i)16-s + (−0.983 + 0.178i)17-s − 18-s + (−0.809 + 0.587i)19-s + (−0.983 − 0.178i)22-s + (−0.134 − 0.990i)23-s + (−0.222 + 0.974i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.995 + 0.0921i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ -0.995 + 0.0921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07174574319 + 1.553106402i\)
\(L(\frac12)\) \(\approx\) \(0.07174574319 + 1.553106402i\)
\(L(1)\) \(\approx\) \(1.237759445 + 0.7871595306i\)
\(L(1)\) \(\approx\) \(1.237759445 + 0.7871595306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.995 + 0.0896i)T \)
3 \( 1 + (0.0448 + 0.998i)T \)
11 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.858 + 0.512i)T \)
17 \( 1 + (-0.983 + 0.178i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.134 - 0.990i)T \)
29 \( 1 + (0.473 + 0.880i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.134 + 0.990i)T \)
41 \( 1 + (-0.550 + 0.834i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.393 - 0.919i)T \)
53 \( 1 + (-0.983 - 0.178i)T \)
59 \( 1 + (0.936 + 0.351i)T \)
61 \( 1 + (-0.691 + 0.722i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.983 + 0.178i)T \)
73 \( 1 + (-0.858 - 0.512i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.393 + 0.919i)T \)
89 \( 1 + (0.858 + 0.512i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−20.69073353479507091720453117952, −20.03428202593791175948758578450, −19.39500943146580041817908918754, −18.64537126706507863693790993433, −17.54454691866988580783000786648, −17.1427764194729732108738335415, −15.77358563233840975821489177664, −15.33037505985970945309946680639, −14.38306275175008476433966166177, −13.61293658432610619381872865572, −12.95296333591963961337132449728, −12.5150453187225726192576526965, −11.51030796282574293495047702007, −10.919226653083936036382098762656, −9.91820624099152035706750121946, −8.664839022878898956155141617, −7.62032135188756615051674139575, −7.19753940287423210194552006122, −6.20564410986171329577339378509, −5.43485409494405003217240323804, −4.64751159948007966571267034535, −3.43249212194512753331895793477, −2.39990381559595609468830634063, −2.02325001150792050033656266079, −0.36719071688378449075950342356, 1.98078393148256235542312699665, 2.768759060937147043643249516571, 3.662642936739552079890093018104, 4.724083377948937630777740530776, 4.945895618778147333126621906920, 6.122253981542466532286685153455, 6.869474820713864350220528449415, 8.05597521864779566404839690313, 8.7625773632946200941692081080, 10.051235843557424265279143407860, 10.56448770567996219528255950780, 11.32946854928524611213839666057, 12.24224298591950810087048953545, 13.00733065284581285770024300525, 13.897329865619040590662119290899, 14.74439454020215548362581191995, 15.10673821663071587967110397343, 16.10852329892920991280963934822, 16.52595302548828695975413591987, 17.35447806781744130835201878197, 18.49129086378015887626901185823, 19.61849457530470818503158683007, 20.1999337218405749412136660625, 20.91771514370436166180950215485, 21.7230857569503687881539722360

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