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Properties

Label 1-3925-3925.1027-r0-0-0
Degree $1$
Conductor $3925$
Sign $0.967 + 0.252i$
Analytic cond. $18.2276$
Root an. cond. $18.2276$
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  + (0.779 + 0.626i)2-s + (0.813 − 0.581i)3-s + (0.215 + 0.976i)4-s + (0.998 + 0.0563i)6-s + (0.748 − 0.663i)7-s + (−0.443 + 0.896i)8-s + (0.324 − 0.945i)9-s + (0.541 + 0.840i)11-s + (0.743 + 0.669i)12-s + (0.207 − 0.978i)13-s + (0.998 − 0.0483i)14-s + (−0.906 + 0.421i)16-s + (0.520 + 0.853i)17-s + (0.845 − 0.534i)18-s + (−0.997 + 0.0643i)19-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.813 − 0.581i)3-s + (0.215 + 0.976i)4-s + (0.998 + 0.0563i)6-s + (0.748 − 0.663i)7-s + (−0.443 + 0.896i)8-s + (0.324 − 0.945i)9-s + (0.541 + 0.840i)11-s + (0.743 + 0.669i)12-s + (0.207 − 0.978i)13-s + (0.998 − 0.0483i)14-s + (−0.906 + 0.421i)16-s + (0.520 + 0.853i)17-s + (0.845 − 0.534i)18-s + (−0.997 + 0.0643i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3925\)    =    \(5^{2} \cdot 157\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(18.2276\)
Root analytic conductor: \(18.2276\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3925} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3925,\ (0:\ ),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.579284157 + 0.5884441315i\)
\(L(\frac12)\) \(\approx\) \(4.579284157 + 0.5884441315i\)
\(L(1)\) \(\approx\) \(2.383166573 + 0.3615256674i\)
\(L(1)\) \(\approx\) \(2.383166573 + 0.3615256674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
157 \( 1 \)
good2 \( 1 + (0.779 + 0.626i)T \)
3 \( 1 + (0.813 - 0.581i)T \)
7 \( 1 + (0.748 - 0.663i)T \)
11 \( 1 + (0.541 + 0.840i)T \)
13 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.520 + 0.853i)T \)
19 \( 1 + (-0.997 + 0.0643i)T \)
23 \( 1 + (0.981 + 0.192i)T \)
29 \( 1 + (0.999 + 0.0241i)T \)
31 \( 1 + (-0.339 - 0.940i)T \)
37 \( 1 + (0.923 - 0.384i)T \)
41 \( 1 + (-0.192 - 0.981i)T \)
43 \( 1 + (-0.948 + 0.316i)T \)
47 \( 1 + (0.982 + 0.184i)T \)
53 \( 1 + (0.619 - 0.784i)T \)
59 \( 1 + (-0.985 - 0.168i)T \)
61 \( 1 + (0.638 + 0.769i)T \)
67 \( 1 + (-0.896 - 0.443i)T \)
71 \( 1 + (0.619 - 0.784i)T \)
73 \( 1 + (-0.594 + 0.804i)T \)
79 \( 1 + (0.964 + 0.262i)T \)
83 \( 1 + (0.339 + 0.940i)T \)
89 \( 1 + (-0.899 + 0.435i)T \)
97 \( 1 + (-0.184 + 0.982i)T \)
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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

−18.78782547235659882835608817031, −18.17890378499741201941224095278, −16.83758093816375892856321448877, −16.34036472141751362874174149102, −15.46370514249137573423474366383, −14.92701485727311961215937595204, −14.27007330526412431227258312705, −13.90945636084240943765797234518, −13.17009380809263088029164282802, −12.23201065188185185797462946451, −11.579757096179553454356716831211, −11.015949349526122526656355136515, −10.33316488345512086895472029572, −9.392402507181875852607617159936, −8.8806452038803662632716345186, −8.32670949577202715614279545535, −7.15096603398066118115561348095, −6.3580646275032084683545414302, −5.47071167297140801182224611680, −4.67656080220382383494643273199, −4.292183657191461801934592732434, −3.226474070947618511638002319877, −2.776494794895674991295285202375, −1.87078419763870636372407424663, −1.13434784797345867641264869627, 0.98500405284758959055638068751, 1.91645861246196759238613594779, 2.67973351477341602133148865176, 3.70270275003824519251082449162, 4.10443672848549846975149378828, 4.98118052753935198246431818347, 5.916802075155124061647917921054, 6.692883040991537136538191631694, 7.302705740032493450383626482048, 7.9815391364804884198763346395, 8.40236133948205717454005627635, 9.27225535967545364892835782097, 10.30664809485513527216807707492, 11.05471592592920533962834310863, 12.047758867230312045352152653921, 12.578724788164425755951667665848, 13.2019092199953519241743781539, 13.766401084778826542076194142592, 14.62682339411891761745563179410, 14.95792751276417645398428903068, 15.36250285833205918699624706813, 16.59891978895177352371925686566, 17.2260036228692754123364956642, 17.67615549015753521906534142732, 18.383286916617537491791627128178

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