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Properties

Label 1-812-812.471-r1-0-0
Degree $1$
Conductor $812$
Sign $0.986 - 0.164i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (0.988 − 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)15-s + (−0.5 + 0.866i)17-s + (0.988 + 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (0.900 − 0.433i)27-s + (−0.0747 + 0.997i)31-s + (−0.988 − 0.149i)33-s + (0.955 − 0.294i)37-s + (−0.365 − 0.930i)39-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)3-s + (0.826 + 0.563i)5-s + (0.955 − 0.294i)9-s + (−0.955 − 0.294i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)15-s + (−0.5 + 0.866i)17-s + (0.988 + 0.149i)19-s + (−0.0747 − 0.997i)23-s + (0.365 + 0.930i)25-s + (0.900 − 0.433i)27-s + (−0.0747 + 0.997i)31-s + (−0.988 − 0.149i)33-s + (0.955 − 0.294i)37-s + (−0.365 − 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.815686899 - 0.3153273343i\)
\(L(\frac12)\) \(\approx\) \(3.815686899 - 0.3153273343i\)
\(L(1)\) \(\approx\) \(1.782927932 - 0.04002879594i\)
\(L(1)\) \(\approx\) \(1.782927932 - 0.04002879594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.988 - 0.149i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.988 + 0.149i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (-0.0747 + 0.997i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.733 - 0.680i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.733 + 0.680i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (-0.955 + 0.294i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.7387219794955739393808024947, −21.18601799687991920446585886416, −20.44991700867037090679966712100, −19.91172242607643535441366284896, −18.79320883193713631480253630960, −18.17059546342553725385722800948, −17.26847783508162608654983105284, −16.17338460506922734877666349271, −15.70844733862399296230265816842, −14.62740033691540477738503132109, −13.72982650449731059940004767991, −13.41358827664182169706726992585, −12.45533966432090341767458464103, −11.365871682260746452734588877341, −10.19667536288936351957369078509, −9.37556165630588193241867913572, −9.096408879223288378452666017931, −7.78996201536556454016409996189, −7.22036192653906767317845799424, −5.88249940056625273507635658660, −4.9084214948517474945848973165, −4.14717901091015222398201750191, −2.736756738552064314517301882579, −2.17018925762037607401353736480, −0.98419728198034136497471761204, 0.86982887477759512086616925398, 2.20473805920493138660151046512, 2.78039681827465718573036306141, 3.71713355073687006352515075913, 5.06376862100399764719012166134, 5.99177515125734637789089421564, 6.99941827740043358557931734332, 7.8775330301674720830300889699, 8.61169445087740026246546922368, 9.677187074467008730047241095073, 10.310656538269740925199776738796, 11.055062260200080920372071406592, 12.70788036310199459952421778677, 12.96673812690162136892689966961, 14.02521101496631233450015357568, 14.53350407519906265048472199098, 15.43007214814175853281249686776, 16.15131418682758901815332834964, 17.47335062510647449978649974985, 18.12077261284394651373789695864, 18.71098729032911686539177183450, 19.68219869570045226576762235206, 20.41874532690102577270195298059, 21.16771277883806982354203177202, 21.85862127876517959362285551718

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