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Properties

Label 1-4235-4235.1157-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.140 + 0.990i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.730 + 0.683i)2-s + (0.743 − 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (0.633 + 0.774i)8-s + (0.104 − 0.994i)9-s + (−0.618 − 0.786i)12-s + (−0.884 − 0.466i)13-s + (−0.991 − 0.132i)16-s + (−0.318 + 0.948i)17-s + (0.603 + 0.797i)18-s + (−0.595 − 0.803i)19-s + (0.690 − 0.723i)23-s + (0.988 + 0.151i)24-s + (0.964 − 0.263i)26-s + (−0.587 − 0.809i)27-s + ⋯
L(s)  = 1  + (−0.730 + 0.683i)2-s + (0.743 − 0.669i)3-s + (0.0665 − 0.997i)4-s + (−0.0855 + 0.996i)6-s + (0.633 + 0.774i)8-s + (0.104 − 0.994i)9-s + (−0.618 − 0.786i)12-s + (−0.884 − 0.466i)13-s + (−0.991 − 0.132i)16-s + (−0.318 + 0.948i)17-s + (0.603 + 0.797i)18-s + (−0.595 − 0.803i)19-s + (0.690 − 0.723i)23-s + (0.988 + 0.151i)24-s + (0.964 − 0.263i)26-s + (−0.587 − 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6300489806 + 0.5468307283i\)
\(L(\frac12)\) \(\approx\) \(0.6300489806 + 0.5468307283i\)
\(L(1)\) \(\approx\) \(0.8039614419 + 0.05378125201i\)
\(L(1)\) \(\approx\) \(0.8039614419 + 0.05378125201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.730 + 0.683i)T \)
3 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (-0.884 - 0.466i)T \)
17 \( 1 + (-0.318 + 0.948i)T \)
19 \( 1 + (-0.595 - 0.803i)T \)
23 \( 1 + (0.690 - 0.723i)T \)
29 \( 1 + (-0.736 + 0.676i)T \)
31 \( 1 + (0.272 + 0.962i)T \)
37 \( 1 + (-0.768 + 0.640i)T \)
41 \( 1 + (-0.516 - 0.856i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (-0.603 + 0.797i)T \)
53 \( 1 + (0.132 + 0.991i)T \)
59 \( 1 + (-0.483 - 0.875i)T \)
61 \( 1 + (0.290 + 0.956i)T \)
67 \( 1 + (0.945 - 0.327i)T \)
71 \( 1 + (-0.870 - 0.491i)T \)
73 \( 1 + (0.0760 + 0.997i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (0.336 + 0.941i)T \)
89 \( 1 + (0.995 - 0.0950i)T \)
97 \( 1 + (0.931 - 0.362i)T \)
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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

−18.473137419265842067430011827368, −17.49363403815757411845921054101, −16.83883849256662888685931817399, −16.407866360400793818791956380398, −15.521727652452219877050214220744, −14.942464863937654632887654963047, −14.11684255839235508573335471424, −13.368298928848897025471574440765, −12.84382942198212048747965436014, −11.661095498090294793742779686384, −11.50714497215215583578814016393, −10.35098577710475334550064102865, −10.00784307553234205510242237826, −9.2243746771244019534174016056, −8.84786190393826326032210021872, −7.872805341074702706033192283, −7.448989067950119016558092774748, −6.57494774323598024139209466167, −5.249332041420173722050519418120, −4.569408456000412482024080525175, −3.75501947433378045891128778951, −3.161344524867212215642687796874, −2.179753658787850914735162011089, −1.839930917232341631142245960819, −0.29253648788339061145401203018, 0.90622303033795073060044965270, 1.77784633402557134791528420302, 2.50486065851338530130447308403, 3.37008123969397248990029103148, 4.55995522195364740962616979413, 5.23212603262427805147967540122, 6.32137085753782548036466406550, 6.76882049109327197891211020172, 7.43194231832899272498648301363, 8.16815134665035622096530914943, 8.75360153173088995413053158503, 9.25453897342892457444892565083, 10.19857741730965264189457320615, 10.72336986483260600962181949374, 11.67369661341882667163186554016, 12.68453598801523831220588188759, 13.03446713227521327465560451822, 14.04868055944617623875860157992, 14.53990528163935506685449773348, 15.23538611249252264056298322632, 15.5692716436363202080584545026, 16.74681468127911275220779629607, 17.24721347167860182968261097010, 17.8145675341694947888393598479, 18.52193703076009331742115006259

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