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Properties

Label 1-43e2-1849.99-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.999 + 0.00339i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
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  + (0.573 + 0.819i)2-s + (0.358 − 0.933i)3-s + (−0.342 + 0.939i)4-s + (0.988 + 0.152i)5-s + (0.970 − 0.241i)6-s + (0.925 + 0.379i)7-s + (−0.966 + 0.257i)8-s + (−0.742 − 0.669i)9-s + (0.441 + 0.897i)10-s + (−0.475 − 0.879i)11-s + (0.753 + 0.656i)12-s + (0.0191 − 0.999i)13-s + (0.219 + 0.975i)14-s + (0.496 − 0.867i)15-s + (−0.765 − 0.643i)16-s + (0.990 + 0.138i)17-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)2-s + (0.358 − 0.933i)3-s + (−0.342 + 0.939i)4-s + (0.988 + 0.152i)5-s + (0.970 − 0.241i)6-s + (0.925 + 0.379i)7-s + (−0.966 + 0.257i)8-s + (−0.742 − 0.669i)9-s + (0.441 + 0.897i)10-s + (−0.475 − 0.879i)11-s + (0.753 + 0.656i)12-s + (0.0191 − 0.999i)13-s + (0.219 + 0.975i)14-s + (0.496 − 0.867i)15-s + (−0.765 − 0.643i)16-s + (0.990 + 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.999 + 0.00339i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.999 + 0.00339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.074227660 + 0.005223353340i\)
\(L(\frac12)\) \(\approx\) \(3.074227660 + 0.005223353340i\)
\(L(1)\) \(\approx\) \(1.869268010 + 0.2290779939i\)
\(L(1)\) \(\approx\) \(1.869268010 + 0.2290779939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.573 + 0.819i)T \)
3 \( 1 + (0.358 - 0.933i)T \)
5 \( 1 + (0.988 + 0.152i)T \)
7 \( 1 + (0.925 + 0.379i)T \)
11 \( 1 + (-0.475 - 0.879i)T \)
13 \( 1 + (0.0191 - 0.999i)T \)
17 \( 1 + (0.990 + 0.138i)T \)
19 \( 1 + (-0.988 + 0.149i)T \)
23 \( 1 + (0.993 - 0.118i)T \)
29 \( 1 + (0.612 + 0.790i)T \)
31 \( 1 + (-0.979 - 0.203i)T \)
37 \( 1 + (0.299 - 0.954i)T \)
41 \( 1 + (-0.512 - 0.858i)T \)
47 \( 1 + (0.951 + 0.308i)T \)
53 \( 1 + (0.706 - 0.707i)T \)
59 \( 1 + (0.293 - 0.956i)T \)
61 \( 1 + (-0.998 + 0.0591i)T \)
67 \( 1 + (0.935 + 0.353i)T \)
71 \( 1 + (0.818 - 0.574i)T \)
73 \( 1 + (0.198 + 0.980i)T \)
79 \( 1 + (0.205 - 0.978i)T \)
83 \( 1 + (0.550 + 0.834i)T \)
89 \( 1 + (-0.967 + 0.251i)T \)
97 \( 1 + (-0.995 + 0.0937i)T \)
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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

−20.519332840297121568274055188008, −19.69800896272073562110144251517, −18.75667772334681564388578550395, −18.056257085754316301450483374808, −17.06209140348751510544526641523, −16.66919335846559310798355551752, −15.23516303579562911199794280111, −14.92421973361689320812213171070, −14.07064664670205873247656696146, −13.68293396939450915278356307945, −12.78007352370468972274753025814, −11.85869498877292540041498433112, −11.01744256886296690446094038703, −10.39479201677104886530088978438, −9.8063358960033752464600561073, −9.11879281058965232494043513451, −8.33820822223041485210848760373, −7.09957784631480639415409894648, −5.98691922797863738814518839915, −5.06282183960962805223946293922, −4.69298639220966940560605105601, −3.92146045457656566819629993802, −2.753492061977979302226260760247, −2.09862760705809245859140022848, −1.2664489788522995473030229620, 0.89198608534998712073198462746, 2.11924253488389510923075345092, 2.83377207788099176860404394965, 3.662517963322430290030742036177, 5.25229730733090860875946600235, 5.48048720867905682891129551642, 6.27218939166221791590028302608, 7.16301018976406563060117444381, 7.94482946260186667988186465146, 8.54981452360961452674647668529, 9.148699352858214512276576853689, 10.53555093938878338635059338470, 11.258027265307863380289306799678, 12.54229702566530000515409356554, 12.70174894534096330835309642259, 13.62139748546854273069385385428, 14.25882438462263781924273210781, 14.721681197504350861103834932596, 15.4464049064795393599364891301, 16.682265835388084286558947494731, 17.188978638263353894918328797261, 17.96308714284758643297407775658, 18.39490890181394592699583217603, 19.109887692033567965336372187828, 20.41249841206926507391207156797

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