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Properties

Label 1-4025-4025.698-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.0119 - 0.999i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
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  + (0.0760 − 0.997i)2-s + (0.956 + 0.290i)3-s + (−0.988 − 0.151i)4-s + (0.362 − 0.931i)6-s + (−0.226 + 0.974i)8-s + (0.830 + 0.556i)9-s + (−0.851 − 0.524i)11-s + (−0.901 − 0.432i)12-s + (0.113 − 0.993i)13-s + (0.953 + 0.299i)16-s + (0.780 − 0.625i)17-s + (0.618 − 0.786i)18-s + (0.548 + 0.836i)19-s + (−0.587 + 0.809i)22-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.0760 − 0.997i)2-s + (0.956 + 0.290i)3-s + (−0.988 − 0.151i)4-s + (0.362 − 0.931i)6-s + (−0.226 + 0.974i)8-s + (0.830 + 0.556i)9-s + (−0.851 − 0.524i)11-s + (−0.901 − 0.432i)12-s + (0.113 − 0.993i)13-s + (0.953 + 0.299i)16-s + (0.780 − 0.625i)17-s + (0.618 − 0.786i)18-s + (0.548 + 0.836i)19-s + (−0.587 + 0.809i)22-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0119 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0119 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.0119 - 0.999i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (698, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.0119 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.668516477 - 1.688508156i\)
\(L(\frac12)\) \(\approx\) \(1.668516477 - 1.688508156i\)
\(L(1)\) \(\approx\) \(1.257987376 - 0.6555686924i\)
\(L(1)\) \(\approx\) \(1.257987376 - 0.6555686924i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0760 - 0.997i)T \)
3 \( 1 + (0.956 + 0.290i)T \)
11 \( 1 + (-0.851 - 0.524i)T \)
13 \( 1 + (0.113 - 0.993i)T \)
17 \( 1 + (0.780 - 0.625i)T \)
19 \( 1 + (0.548 + 0.836i)T \)
29 \( 1 + (0.998 + 0.0570i)T \)
31 \( 1 + (0.683 - 0.730i)T \)
37 \( 1 + (-0.556 + 0.830i)T \)
41 \( 1 + (0.985 - 0.170i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (-0.662 + 0.749i)T \)
59 \( 1 + (-0.595 + 0.803i)T \)
61 \( 1 + (-0.797 - 0.603i)T \)
67 \( 1 + (0.983 + 0.179i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.986 - 0.161i)T \)
79 \( 1 + (-0.861 - 0.508i)T \)
83 \( 1 + (0.717 - 0.696i)T \)
89 \( 1 + (0.483 - 0.875i)T \)
97 \( 1 + (0.717 + 0.696i)T \)
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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

−18.46978860953941915216395663210, −17.98826616238492774047480351443, −17.371315214266368234849949969943, −16.35958911505105310260522222706, −15.843337721206583053723919519153, −15.261936019933662459577877386413, −14.51339616643211511025661852616, −14.00001496563874007522439384990, −13.44943937977243027306993664630, −12.62287527968203452569743101518, −12.22840178310397219612100680157, −10.97957824613723257755689115833, −9.89654693697758617459022300526, −9.58316019743023684126093579860, −8.67987234622845157735712896827, −8.11907730205686336282253010741, −7.509424163908634783332607095294, −6.7903181326906172345697754818, −6.24196726571305245609922236725, −5.053638454375684788377840660726, −4.59182455735021004480406272867, −3.62472525601385185113041359157, −2.94840973892176574676182284554, −1.924054348776229132717773416281, −0.93877765098110719987594933102, 0.70047331248343589700580294279, 1.58496059579440194022337831024, 2.63601243906933994821702509963, 3.093312477763947226082237089118, 3.64208841953062525167462049527, 4.70296651051884910906375006454, 5.24494069390376159869381388671, 6.100269949522677157467615536292, 7.53340237875239560967916210926, 8.04859120245305748047109569998, 8.55162286736392392030629271211, 9.505830182145844991304125666649, 10.10409023325721470617737630932, 10.47854426999299404772584468992, 11.3742962645828356400787565231, 12.24614758249766005494296541884, 12.811107154888930307734564803821, 13.67391658156638792568375436091, 13.90552493185210582953376929882, 14.74556686557709793931243902161, 15.5019950235668936566455689927, 16.1048959755267149461669778682, 17.023463090647545060201106770353, 17.93876007226512052085792230955, 18.681714880078846006397775041546

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