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Properties

Label 1-812-812.23-r1-0-0
Degree $1$
Conductor $812$
Sign $-0.604 + 0.796i$
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  + (−0.826 + 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + (−0.5 + 0.866i)17-s + (−0.826 − 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.955 − 0.294i)31-s + (0.826 + 0.563i)33-s + (0.365 − 0.930i)37-s + (−0.0747 + 0.997i)39-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (−0.733 + 0.680i)5-s + (0.365 − 0.930i)9-s + (−0.365 − 0.930i)11-s + (0.623 − 0.781i)13-s + (0.222 − 0.974i)15-s + (−0.5 + 0.866i)17-s + (−0.826 − 0.563i)19-s + (−0.955 + 0.294i)23-s + (0.0747 − 0.997i)25-s + (0.222 + 0.974i)27-s + (−0.955 − 0.294i)31-s + (0.826 + 0.563i)33-s + (0.365 − 0.930i)37-s + (−0.0747 + 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.604 + 0.796i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ -0.604 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1925805729 + 0.3879305579i\)
\(L(\frac12)\) \(\approx\) \(0.1925805729 + 0.3879305579i\)
\(L(1)\) \(\approx\) \(0.6109691665 + 0.1048819133i\)
\(L(1)\) \(\approx\) \(0.6109691665 + 0.1048819133i\)

Euler product

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

−21.85538692604712994390047741373, −20.84825931330324661409212618109, −20.14442163150450915297688443184, −19.27027273132137600191400898822, −18.43623599901519309380708585325, −17.838230846968046158022119464104, −16.79469853133897629153209678786, −16.236861065422450506404511615845, −15.547634045946904545002381993655, −14.39909037351179816666810235044, −13.31172119122204261718521658561, −12.660828152704930902549913765386, −11.93284533459945888195268770720, −11.282154601932859989462339765767, −10.34982479521299849924878491191, −9.23748182407340856714040038362, −8.24351086478954876639586449399, −7.43920121934246021729997465581, −6.64118791593643889986839238147, −5.63553474972288508657785486244, −4.6153437791289826976048305172, −4.06473103675278483600872440532, −2.327096168792506863074587509531, −1.36848330273667108334205671175, −0.1675910863949312598136110546, 0.70528579364176598154453339707, 2.4596420434415499734376008310, 3.71666245009398318240934516438, 4.11888319850184823352800293540, 5.577947974223020463539007646110, 6.09343170217812947516546526294, 7.132240279948895308322731906617, 8.16058263148297404050847596494, 9.00379892287334772089140280785, 10.35209330635701477328150012402, 10.82738686747322918032069306512, 11.35799095691726555328944136180, 12.4106239011243767344029735193, 13.21616340611963759557700784389, 14.41101347306453665891426976707, 15.30133010417097848064149705281, 15.77982119041340276265694446883, 16.54151536497249734730214805397, 17.58094905381142459447997956907, 18.18365056550471384709351017881, 19.057685408105389653488635880062, 19.88202405973071447953208394251, 20.83156512284934947560061908708, 21.86935231003756948842863104936, 22.08519248203468849407412483907

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