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Properties

Label 1-812-812.51-r1-0-0
Degree $1$
Conductor $812$
Sign $-0.799 - 0.601i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.799 - 0.601i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ -0.799 - 0.601i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1558354382 - 0.4664121535i\)
\(L(\frac12)\) \(\approx\) \(0.1558354382 - 0.4664121535i\)
\(L(1)\) \(\approx\) \(0.8198278747 + 0.006356008739i\)
\(L(1)\) \(\approx\) \(0.8198278747 + 0.006356008739i\)

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

−22.17003634714836981200357139040, −21.48014621167595615583934453864, −21.22997827501073372155176588340, −19.777998934676991018769763618624, −19.135204707662800558886634274723, −18.19077222452700048996399827904, −17.24637427330324944494767266127, −16.92686016084669263967618190163, −16.25929960036928622217465387601, −15.08599144558639533452532687721, −14.07576214730531707913935891311, −13.34446217145436382810318818178, −12.36480729692755199088784586108, −11.881759838475161010245552323486, −10.82646987501170492945985208513, −10.01964134706044673174417871590, −9.1856190649771899902612262791, −8.2838083419890127366036103012, −6.88459902872294734866747365662, −6.333961176762971478148541521568, −5.47735961924563113061741085629, −4.61548693231450455146435867545, −3.64327486201707117996173665720, −1.802272846948377334567210212025, −1.39235333905559225638131489973, 0.12386831344812600006931317190, 1.34019451265072274498330152508, 2.48874868856449713808025622299, 3.709185590903733997070110628711, 4.85026725113815546147308585804, 5.61847178968723910083117385503, 6.55893712107154699227859931734, 7.03112057744706380218394748006, 8.40195906656099230800261299733, 9.664513343979317211675749034471, 10.104503199942107532435539266829, 10.96968497020187775938022403890, 11.81100631782120833921036960480, 12.65315835292740419183603217750, 13.46488624227166270672254897233, 14.57807763902686406125786704716, 15.124576348470050106359188015881, 16.24890195815007706158326536, 17.12955994987080210534129105744, 17.492039065493688996378479276799, 18.40895628519290153512382567798, 19.030539232013679840938683140, 20.31149198542435615593778966465, 21.02236637884526625189396960534, 21.954955478252055628889348243120

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