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Properties

Label 1-1925-1925.114-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.204 + 0.978i$
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  + (0.5 + 0.866i)2-s + (0.978 + 0.207i)3-s + (−0.5 + 0.866i)4-s + (0.309 + 0.951i)6-s − 8-s + (0.913 + 0.406i)9-s + (−0.669 + 0.743i)12-s + (0.809 − 0.587i)13-s + (−0.5 − 0.866i)16-s + (0.978 + 0.207i)17-s + (0.104 + 0.994i)18-s + (−0.5 − 0.866i)19-s + (0.978 − 0.207i)23-s + (−0.978 − 0.207i)24-s + (0.913 + 0.406i)26-s + (0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.978 + 0.207i)3-s + (−0.5 + 0.866i)4-s + (0.309 + 0.951i)6-s − 8-s + (0.913 + 0.406i)9-s + (−0.669 + 0.743i)12-s + (0.809 − 0.587i)13-s + (−0.5 − 0.866i)16-s + (0.978 + 0.207i)17-s + (0.104 + 0.994i)18-s + (−0.5 − 0.866i)19-s + (0.978 − 0.207i)23-s + (−0.978 − 0.207i)24-s + (0.913 + 0.406i)26-s + (0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.506164015 + 2.036460113i\)
\(L(\frac12)\) \(\approx\) \(2.506164015 + 2.036460113i\)
\(L(1)\) \(\approx\) \(1.674828429 + 0.9746182491i\)
\(L(1)\) \(\approx\) \(1.674828429 + 0.9746182491i\)

Euler product

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

−19.88252285808007955852176464137, −19.26374166578892960515736115102, −18.6442385763745568733838978393, −18.18868187683708065146042519284, −16.99072571266028795143527407666, −16.01237262931868832384116053620, −15.16277244371065766147551270481, −14.55766900845511917993643194155, −13.83947137662653678057830872403, −13.3644650749510242505440136444, −12.49153037429969400913239769893, −11.90524189434343977471619112251, −10.98626513722998022772796929447, −10.058542400481902744872347857545, −9.597257789509393731650085134708, −8.576825261166236461606943288458, −8.13306752289107877028488505922, −6.8202291392528209884834326491, −6.18189357930144829267999525541, −4.994294940174791822315243602814, −4.23970574056399989706933878009, −3.32660187308558987436060661533, −2.86933861132476467802416766930, −1.66519399484668121653332744054, −1.152276464315368946491550568457, 1.03287405293192254665239195572, 2.53247119071864584084168970634, 3.18416144378252869691558180951, 3.99431063164985132552497963255, 4.78197534950308279156430624728, 5.640021183148992121344151050992, 6.62293098297744339574220337652, 7.31567378898736174852050106574, 8.26740165737417087383752995055, 8.56623769996999493592122015698, 9.5011844905576680992616630145, 10.33409734938932729713277575496, 11.33064480741768976184083795449, 12.46883587274670526233057857381, 13.06699778783184844525339483412, 13.64796439942988855240685461125, 14.48574327078283035733920901463, 14.95811468458258564749963125200, 15.77615108592740032125825080389, 16.18056800801172134820797336756, 17.23051412210093420461038724480, 17.853994732428110735059942701373, 18.81930919817363146701320848260, 19.36731973722465712372448184672, 20.41669352712329560258796760048

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