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Properties

Label 1-925-925.16-r0-0-0
Degree $1$
Conductor $925$
Sign $-0.0367 + 0.999i$
Analytic cond. $4.29568$
Root an. cond. $4.29568$
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  + (−0.374 + 0.927i)2-s + (−0.241 + 0.970i)3-s + (−0.719 − 0.694i)4-s + (−0.809 − 0.587i)6-s + (−0.939 − 0.342i)7-s + (0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (−0.978 − 0.207i)11-s + (0.848 − 0.529i)12-s + (0.990 + 0.139i)13-s + (0.669 − 0.743i)14-s + (0.0348 + 0.999i)16-s + (−0.719 + 0.694i)17-s + (0.766 − 0.642i)18-s + (−0.997 + 0.0697i)19-s + ⋯
L(s)  = 1  + (−0.374 + 0.927i)2-s + (−0.241 + 0.970i)3-s + (−0.719 − 0.694i)4-s + (−0.809 − 0.587i)6-s + (−0.939 − 0.342i)7-s + (0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (−0.978 − 0.207i)11-s + (0.848 − 0.529i)12-s + (0.990 + 0.139i)13-s + (0.669 − 0.743i)14-s + (0.0348 + 0.999i)16-s + (−0.719 + 0.694i)17-s + (0.766 − 0.642i)18-s + (−0.997 + 0.0697i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.0367 + 0.999i$
Analytic conductor: \(4.29568\)
Root analytic conductor: \(4.29568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 925,\ (0:\ ),\ -0.0367 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4467266838 + 0.4634774610i\)
\(L(\frac12)\) \(\approx\) \(0.4467266838 + 0.4634774610i\)
\(L(1)\) \(\approx\) \(0.5097724951 + 0.3553560916i\)
\(L(1)\) \(\approx\) \(0.5097724951 + 0.3553560916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.374 + 0.927i)T \)
3 \( 1 + (-0.241 + 0.970i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.990 + 0.139i)T \)
17 \( 1 + (-0.719 + 0.694i)T \)
19 \( 1 + (-0.997 + 0.0697i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.559 - 0.829i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (0.559 + 0.829i)T \)
71 \( 1 + (-0.241 + 0.970i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (-0.719 + 0.694i)T \)
89 \( 1 + (-0.615 - 0.788i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−21.62234825940911019015751716181, −20.73410239579603446118055917318, −19.89453528082363139479407387583, −19.29129022324301041667331127638, −18.418058154245149811850175687061, −18.15669139622197185041358882104, −17.18942186211274307426919710529, −16.29550216638185908561240879757, −15.46231988192245087250032084720, −14.017480558924104254158992504602, −13.25050550835025518028725230905, −12.79616301356619699424899877049, −12.153941045245433386409163895741, −10.90649825349345224861399990427, −10.75062638626114209605779618462, −9.25974310472783946032479921899, −8.78955766560136697046025244143, −7.71569682776344496073150306471, −6.9375793656711181990624503757, −5.87225713244022230157548580030, −4.92231426550957840515588866354, −3.53224694963674703869552615535, −2.69970726134731804436298312248, −1.89697806986779626676524053327, −0.612999411695432378950883655382, 0.60873704873071253402858671047, 2.53070141572301243885167339077, 3.920410718029057270550277646676, 4.379928351512257632525115573070, 5.753836735539949842271569711803, 6.09741438574715399309939868962, 7.13234349913462284893847424554, 8.343734125987756840886573174207, 8.91748156880567193034080042890, 9.80514383273508183004201803589, 10.65286826738104127702244421726, 11.015621281166881614034801903678, 12.77756789465011465962999632706, 13.31083641102038209470841844386, 14.33469265893290417557892460816, 15.29960109514741863345076383036, 15.708210599461786042260930639315, 16.5037336806988105070694411355, 17.0189478476894409542496128330, 17.93174753494106042045799742654, 18.84956586075908098731594906573, 19.53847428012999860923679030267, 20.57962028908355733743680874924, 21.32510610953526574556175620713, 22.305454384421115749518781727321

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