↑

Properties

Label 1-805-805.339-r0-0-0
Degree $1$
Conductor $805$
Sign $-0.207 + 0.978i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
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  + (−0.981 + 0.189i)2-s + (0.0475 − 0.998i)3-s + (0.928 − 0.371i)4-s + (0.142 + 0.989i)6-s + (−0.841 + 0.540i)8-s + (−0.995 − 0.0950i)9-s + (−0.981 − 0.189i)11-s + (−0.327 − 0.945i)12-s + (−0.959 + 0.281i)13-s + (0.723 − 0.690i)16-s + (0.786 − 0.618i)17-s + (0.995 − 0.0950i)18-s + (−0.786 − 0.618i)19-s + 22-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.0475 − 0.998i)3-s + (0.928 − 0.371i)4-s + (0.142 + 0.989i)6-s + (−0.841 + 0.540i)8-s + (−0.995 − 0.0950i)9-s + (−0.981 − 0.189i)11-s + (−0.327 − 0.945i)12-s + (−0.959 + 0.281i)13-s + (0.723 − 0.690i)16-s + (0.786 − 0.618i)17-s + (0.995 − 0.0950i)18-s + (−0.786 − 0.618i)19-s + 22-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ -0.207 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08533007056 + 0.1052907057i\)
\(L(\frac12)\) \(\approx\) \(0.08533007056 + 0.1052907057i\)
\(L(1)\) \(\approx\) \(0.4971071439 - 0.1224834458i\)
\(L(1)\) \(\approx\) \(0.4971071439 - 0.1224834458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
3 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (-0.981 - 0.189i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (0.786 - 0.618i)T \)
19 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (-0.995 - 0.0950i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.235 + 0.971i)T \)
59 \( 1 + (-0.723 - 0.690i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.928 - 0.371i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.888 + 0.458i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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

−21.71998428826210083389883897644, −21.03505881557440763779223798479, −20.53129477038083498115677915321, −19.57401328164119696615956146737, −18.94083180467914634313292792691, −17.91670222291397230170177994423, −17.05388907830695522612099610017, −16.61883156252704564644927706163, −15.560884945919788028764440155232, −15.11298136460737810236191592457, −14.163675920240033717854050751846, −12.704382895853428678285210160337, −12.07117408539197583389045826698, −10.93419396432690164862592441971, −10.24645192467833518200293782930, −9.87787623596884932179051127535, −8.71025231321021890405944351523, −8.098252738815209960422390109648, −7.15692257587849649816043183478, −5.88997324128072011034650561231, −5.04716536716050991316455924739, −3.77906331184980032159686132378, −2.87948220808342708212698006828, −1.91416476316045757751932632282, −0.087678153031515412495966816263, 1.15834439005816348938236146894, 2.38790688180039086800997645550, 2.91004940155229143233550825117, 4.87894781213612306410397530249, 5.86751563398653701131901014960, 6.74863376236800576167018041791, 7.56691366036010207781082806764, 8.11119277870813131223201160878, 9.083709402662487174967831787054, 9.99308216219033124965314705197, 10.92185139961539338434498022468, 11.814568249991674280798240613994, 12.48881141131728891534201971114, 13.53603151965996130897287933354, 14.42632086468634173855612803913, 15.28735668530791516847346088445, 16.22608223291680741179298151565, 17.129506162279984961814420360595, 17.66569547647860550953891767274, 18.54220631200926591186528749727, 19.15095790709084096885153750150, 19.68877887448498882633881555568, 20.725653372363365758030581637562, 21.36477653343707254393599776503, 22.76414482792245398079126803253

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