↑

Properties

Label 1-4805-4805.1028-r1-0-0
Degree $1$
Conductor $4805$
Sign $-0.973 - 0.227i$
Analytic cond. $516.368$
Root an. cond. $516.368$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.937 − 0.347i)2-s + (−0.975 + 0.217i)3-s + (0.758 − 0.651i)4-s + (−0.839 + 0.543i)6-s + (−0.0337 + 0.999i)7-s + (0.485 − 0.874i)8-s + (0.905 − 0.425i)9-s + (−0.972 + 0.234i)11-s + (−0.598 + 0.801i)12-s + (−0.996 + 0.0843i)13-s + (0.315 + 0.948i)14-s + (0.151 − 0.988i)16-s + (0.830 + 0.557i)17-s + (0.701 − 0.713i)18-s + (−0.990 − 0.134i)19-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)2-s + (−0.975 + 0.217i)3-s + (0.758 − 0.651i)4-s + (−0.839 + 0.543i)6-s + (−0.0337 + 0.999i)7-s + (0.485 − 0.874i)8-s + (0.905 − 0.425i)9-s + (−0.972 + 0.234i)11-s + (−0.598 + 0.801i)12-s + (−0.996 + 0.0843i)13-s + (0.315 + 0.948i)14-s + (0.151 − 0.988i)16-s + (0.830 + 0.557i)17-s + (0.701 − 0.713i)18-s + (−0.990 − 0.134i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4805\)    =    \(5 \cdot 31^{2}\)
Sign: $-0.973 - 0.227i$
Analytic conductor: \(516.368\)
Root analytic conductor: \(516.368\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4805} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4805,\ (1:\ ),\ -0.973 - 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01128923679 + 0.09776980525i\)
\(L(\frac12)\) \(\approx\) \(0.01128923679 + 0.09776980525i\)
\(L(1)\) \(\approx\) \(1.149380854 + 0.01567148716i\)
\(L(1)\) \(\approx\) \(1.149380854 + 0.01567148716i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.937 - 0.347i)T \)
3 \( 1 + (-0.975 + 0.217i)T \)
7 \( 1 + (-0.0337 + 0.999i)T \)
11 \( 1 + (-0.972 + 0.234i)T \)
13 \( 1 + (-0.996 + 0.0843i)T \)
17 \( 1 + (0.830 + 0.557i)T \)
19 \( 1 + (-0.990 - 0.134i)T \)
23 \( 1 + (0.968 - 0.250i)T \)
29 \( 1 + (0.250 + 0.968i)T \)
37 \( 1 + (-0.425 + 0.905i)T \)
41 \( 1 + (0.857 - 0.514i)T \)
43 \( 1 + (-0.912 + 0.409i)T \)
47 \( 1 + (-0.968 + 0.250i)T \)
53 \( 1 + (0.676 + 0.736i)T \)
59 \( 1 + (0.905 + 0.425i)T \)
61 \( 1 + (0.918 + 0.394i)T \)
67 \( 1 + (0.0675 - 0.997i)T \)
71 \( 1 + (0.990 - 0.134i)T \)
73 \( 1 + (-0.747 - 0.664i)T \)
79 \( 1 + (0.664 + 0.747i)T \)
83 \( 1 + (0.168 - 0.985i)T \)
89 \( 1 + (-0.528 - 0.848i)T \)
97 \( 1 + (-0.651 + 0.758i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.46797530468109179779284552042, −16.67812168863694122424629797832, −16.47558486718384479870104024636, −15.63653457264283000592754721169, −14.91643441925934393022493526999, −14.20957054778663457031579481032, −13.41935680334322053125791003424, −12.9136369006909969317974238946, −12.43584823589741476714539505427, −11.5098850555061798785264709340, −11.09442791310257055516428264950, −10.25974183421078800133726043793, −9.83004196107761563848025667327, −8.28900939687703056362778008569, −7.709681195201104193616109655702, −7.038202130359270828583373909641, −6.62711554764391748444325287883, −5.54080192967838710565854589869, −5.20241600353904698254033717866, −4.46313223284719768626656290213, −3.744281217722316516522847172, −2.76757875134278226811176173309, −1.97151847040619196118017115529, −0.8059186510584637497729254421, −0.0131879110736315402540633511, 1.13518062237439090128896110120, 2.06931371849842978651042857960, 2.76780259423152897016138114119, 3.59406147103488208885347050191, 4.69376438951910397064685449764, 5.02900455597197823822196557388, 5.613515833008003885164330055347, 6.39211999921519462732618139300, 6.99302889798442907411970725286, 7.86799127469682950659477805446, 8.92413073345493941536594895866, 9.87243154125801396061990906479, 10.34122631814413788700107954751, 10.98550385343523073090236182913, 11.722477010976776432365654152161, 12.41985860058429426566416210996, 12.66551736163958572109283291034, 13.33042554320402809218813103336, 14.57736670560078265846256363612, 14.96127963970611580629314517341, 15.474259420206029988087436037115, 16.274411148050543233820123700062, 16.79826570836621209727677946954, 17.638309460530036984615191684763, 18.423501958241798594346667715977

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