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Properties

Label 1-763-763.149-r1-0-0
Degree $1$
Conductor $763$
Sign $-0.999 + 0.0247i$
Analytic cond. $81.9957$
Root an. cond. $81.9957$
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  + (0.984 + 0.173i)2-s + (−0.0581 + 0.998i)3-s + (0.939 + 0.342i)4-s + (−0.993 + 0.116i)5-s + (−0.230 + 0.973i)6-s + (0.866 + 0.5i)8-s + (−0.993 − 0.116i)9-s + (−0.998 − 0.0581i)10-s + (0.549 + 0.835i)11-s + (−0.396 + 0.918i)12-s + (0.116 + 0.993i)13-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + (−0.957 − 0.286i)18-s + (0.342 + 0.939i)19-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (−0.0581 + 0.998i)3-s + (0.939 + 0.342i)4-s + (−0.993 + 0.116i)5-s + (−0.230 + 0.973i)6-s + (0.866 + 0.5i)8-s + (−0.993 − 0.116i)9-s + (−0.998 − 0.0581i)10-s + (0.549 + 0.835i)11-s + (−0.396 + 0.918i)12-s + (0.116 + 0.993i)13-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s + (−0.957 − 0.286i)18-s + (0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(763\)    =    \(7 \cdot 109\)
Sign: $-0.999 + 0.0247i$
Analytic conductor: \(81.9957\)
Root analytic conductor: \(81.9957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{763} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 763,\ (1:\ ),\ -0.999 + 0.0247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03936262484 + 3.175044187i\)
\(L(\frac12)\) \(\approx\) \(0.03936262484 + 3.175044187i\)
\(L(1)\) \(\approx\) \(1.283745504 + 1.139251648i\)
\(L(1)\) \(\approx\) \(1.283745504 + 1.139251648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (-0.0581 + 0.998i)T \)
5 \( 1 + (-0.993 + 0.116i)T \)
11 \( 1 + (0.549 + 0.835i)T \)
13 \( 1 + (0.116 + 0.993i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.0581 - 0.998i)T \)
31 \( 1 + (-0.597 + 0.802i)T \)
37 \( 1 + (0.802 + 0.597i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.957 - 0.286i)T \)
53 \( 1 + (-0.802 + 0.597i)T \)
59 \( 1 + (0.448 - 0.893i)T \)
61 \( 1 + (-0.973 - 0.230i)T \)
67 \( 1 + (-0.802 - 0.597i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.116 + 0.993i)T \)
83 \( 1 + (0.835 + 0.549i)T \)
89 \( 1 + (-0.286 - 0.957i)T \)
97 \( 1 + (0.597 + 0.802i)T \)
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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

−22.10969026178843482453331238485, −20.915148831117000661358458788308, −20.06199285908725765188352520067, −19.55627632286103124873718461169, −18.84697230725781837992416108613, −17.919414070114249318456909656962, −16.57461604949943663883748837873, −16.22252725213553447523420448445, −14.84486462448062611229121211002, −14.55748887259547272067516758221, −13.30948796945198460890598783411, −12.854748905651137890789722211090, −11.977189770646985372038245547355, −11.33746689710111530329769285711, −10.68748134300959567197200752030, −9.070197103421725955241991174125, −7.963469390079599210292179244, −7.384630176384795394344928788349, −6.41786131304609376483194273095, −5.57474962563923288560796051534, −4.616792952713129908369384144917, −3.28061011323772191632781762362, −2.908557445660291823268817160, −1.27306747255283857008109034532, −0.534152890114046516752429931845, 1.553350135404322043973961239555, 3.04744509373185153720781840036, 3.79313254864431004618962873309, 4.40468799212377934186417115042, 5.28130409684260099773142048069, 6.33941456654215066275808090455, 7.328490863619163268694199618236, 8.157263173988343384090699616565, 9.35543229914247708657164515372, 10.2890585013496221059302706369, 11.330586981881937569403285877516, 11.82413184905637517628451737904, 12.54102363611425203112577289907, 13.91405527775603096887162483319, 14.6246203694809804978914146550, 15.1273260393805025031188911440, 16.00537875466744687834970567160, 16.57731024063525964686558389379, 17.31314313831160959308205991715, 18.84077028252966936544175196773, 19.68216210844977989458051946362, 20.40112273080898485478367066195, 21.12306971492477095570309779910, 21.82160008497050082864093523839, 22.76967453948113001961853869320

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