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Properties

Label 1-35e2-1225.71-r0-0-0
Degree $1$
Conductor $1225$
Sign $0.400 - 0.916i$
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  + (−0.963 + 0.266i)2-s + (0.134 − 0.990i)3-s + (0.858 − 0.512i)4-s + (0.134 + 0.990i)6-s + (−0.691 + 0.722i)8-s + (−0.963 − 0.266i)9-s + (−0.963 + 0.266i)11-s + (−0.393 − 0.919i)12-s + (−0.0448 + 0.998i)13-s + (0.473 − 0.880i)16-s + (0.858 + 0.512i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.858 − 0.512i)22-s + (−0.393 + 0.919i)23-s + (0.623 + 0.781i)24-s + ⋯
L(s)  = 1  + (−0.963 + 0.266i)2-s + (0.134 − 0.990i)3-s + (0.858 − 0.512i)4-s + (0.134 + 0.990i)6-s + (−0.691 + 0.722i)8-s + (−0.963 − 0.266i)9-s + (−0.963 + 0.266i)11-s + (−0.393 − 0.919i)12-s + (−0.0448 + 0.998i)13-s + (0.473 − 0.880i)16-s + (0.858 + 0.512i)17-s + 18-s + (0.309 − 0.951i)19-s + (0.858 − 0.512i)22-s + (−0.393 + 0.919i)23-s + (0.623 + 0.781i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(5.68887\)
Root analytic conductor: \(5.68887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (0:\ ),\ 0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6719109409 - 0.4398229367i\)
\(L(\frac12)\) \(\approx\) \(0.6719109409 - 0.4398229367i\)
\(L(1)\) \(\approx\) \(0.6634188058 - 0.1574263095i\)
\(L(1)\) \(\approx\) \(0.6634188058 - 0.1574263095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.963 + 0.266i)T \)
3 \( 1 + (0.134 - 0.990i)T \)
11 \( 1 + (-0.963 + 0.266i)T \)
13 \( 1 + (-0.0448 + 0.998i)T \)
17 \( 1 + (0.858 + 0.512i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.393 + 0.919i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.393 - 0.919i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (0.936 + 0.351i)T \)
53 \( 1 + (0.858 - 0.512i)T \)
59 \( 1 + (0.473 - 0.880i)T \)
61 \( 1 + (0.753 - 0.657i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (-0.0448 - 0.998i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.936 - 0.351i)T \)
89 \( 1 + (-0.0448 - 0.998i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−20.8883137989721016239494200272, −20.619076047133405164089483720316, −19.91282593302260121074631229844, −18.85908565570063854904512932527, −18.31039746689501415600524847180, −17.42643006297181464102022977756, −16.532671009143735907353147714479, −16.13135207584796846092099910266, −15.32045841017491101846855999927, −14.60589453519242679707466953098, −13.52568581593869902983456431878, −12.44512973889169608219714860150, −11.729091371419791941678138720729, −10.6742748481323350933653070521, −10.27265605629832407964084488343, −9.67068299742097359124320094740, −8.54876842973882176500463735311, −8.10641838011492361823968894514, −7.21892698256872033422555955114, −5.83535757339467268932437048336, −5.272431072594142064192597222380, −3.896718286656213618049731884001, −3.08445100773054009532237344752, −2.39103686140005857309309749513, −0.85139162135256371876191722064, 0.56949394770647835510101966106, 1.80167561921247154825135886406, 2.36698254504986559074852762850, 3.54976910065713836642311101719, 5.19140583995561521620108152072, 5.92467763483592614044057026767, 6.85593993927858455595134513894, 7.569779741721536298263350336719, 8.06276729986227585370319818502, 9.126339478764833098785148112435, 9.69986701305006187548088925060, 10.86561449743972138437793684090, 11.51154096785566214719332117511, 12.32112773271144348527202398467, 13.227469592492960988916927556714, 14.076945033957228976693235759933, 14.89015750087551828348790443569, 15.70698673060102435802281896684, 16.59823267472884007273434222879, 17.327086767967020870134623307485, 17.99958986864405327931617499401, 18.68909905125662015848698086522, 19.247735029273091287325163518060, 19.93261158184288455633649336013, 20.780539705051016711543679388

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