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Properties

Label 1-1925-1925.1153-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.989 - 0.143i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (−0.743 − 0.669i)2-s + (0.866 + 0.5i)3-s + (0.104 + 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.406 + 0.913i)12-s − i·13-s + (−0.978 + 0.207i)16-s + (0.207 − 0.978i)17-s + (0.207 − 0.978i)18-s + (−0.104 + 0.994i)19-s + (−0.994 − 0.104i)23-s + (0.913 − 0.406i)24-s + (−0.669 + 0.743i)26-s + i·27-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (0.866 + 0.5i)3-s + (0.104 + 0.994i)4-s + (−0.309 − 0.951i)6-s + (0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.406 + 0.913i)12-s − i·13-s + (−0.978 + 0.207i)16-s + (0.207 − 0.978i)17-s + (0.207 − 0.978i)18-s + (−0.104 + 0.994i)19-s + (−0.994 − 0.104i)23-s + (0.913 − 0.406i)24-s + (−0.669 + 0.743i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.989 - 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.545697848 - 0.1113451649i\)
\(L(\frac12)\) \(\approx\) \(1.545697848 - 0.1113451649i\)
\(L(1)\) \(\approx\) \(1.039759874 - 0.09400119468i\)
\(L(1)\) \(\approx\) \(1.039759874 - 0.09400119468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.743 - 0.669i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.207 - 0.978i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.743 + 0.669i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−19.87770957180559905751695110799, −19.16020959823333484645459204001, −18.7135796725691787384987663160, −17.8670435381631582786699774906, −17.28292219224602390849633321660, −16.37647041786261655051966115591, −15.65449070447052479141924105512, −14.92008170473001006778689788263, −14.30577034050907454240532995979, −13.66114430345314806193342991189, −12.85827403986350619313374972667, −11.843736889806717674444331526680, −11.02961095629589239519166699358, −9.965170613836969263558052625063, −9.44953848954808612597629398027, −8.58196015806167101660916087524, −8.123088328217283421761874158026, −7.226799365522793982608298028991, −6.58542714016975169409923030141, −5.90266887528404399079796422122, −4.646505872149355415774803482482, −3.83297526321309125173499015017, −2.505557018525322704443736967, −1.8480564934693971446203747149, −0.841325482252380610355288855671, 0.85767178737143000523239259799, 2.01220545877974998131726203, 2.77862214050328395805842643511, 3.488510377525492311875386985860, 4.28891550157590232338878618073, 5.25796258454274167619530431924, 6.51748842370096104523654608700, 7.72194005953580079219827186573, 7.96335022230395898148525856047, 8.845101546542651293641570781327, 9.627174129057134511816888070787, 10.23063067813522143189719404318, 10.754023824152592536552839281583, 11.87633006152401522024509796948, 12.472465141198669281385327738428, 13.41747361172056836828772781553, 14.02920494715034924157554745054, 14.924353884578643140334654093441, 15.91246086928837341805104238965, 16.20707413392342255584645776678, 17.20725216565899398619222962523, 18.063446134939983015803164890581, 18.60695541843452428454613454906, 19.48046592754732662604444143989, 19.99519652529347829234665352840

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