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Properties

Label 1-781-781.179-r0-0-0
Degree $1$
Conductor $781$
Sign $0.821 - 0.569i$
Analytic cond. $3.62694$
Root an. cond. $3.62694$
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  + (−0.995 − 0.0896i)2-s + (0.473 − 0.880i)3-s + (0.983 + 0.178i)4-s + (0.309 + 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (−0.963 − 0.266i)8-s + (−0.550 − 0.834i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.983 + 0.178i)15-s + (0.936 + 0.351i)16-s + (0.309 + 0.951i)17-s + (0.473 + 0.880i)18-s + ⋯
L(s)  = 1  + (−0.995 − 0.0896i)2-s + (0.473 − 0.880i)3-s + (0.983 + 0.178i)4-s + (0.309 + 0.951i)5-s + (−0.550 + 0.834i)6-s + (0.753 − 0.657i)7-s + (−0.963 − 0.266i)8-s + (−0.550 − 0.834i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)12-s + (0.936 − 0.351i)13-s + (−0.809 + 0.587i)14-s + (0.983 + 0.178i)15-s + (0.936 + 0.351i)16-s + (0.309 + 0.951i)17-s + (0.473 + 0.880i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 781 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 781 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(781\)    =    \(11 \cdot 71\)
Sign: $0.821 - 0.569i$
Analytic conductor: \(3.62694\)
Root analytic conductor: \(3.62694\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{781} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 781,\ (0:\ ),\ 0.821 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309650043 - 0.4097761015i\)
\(L(\frac12)\) \(\approx\) \(1.309650043 - 0.4097761015i\)
\(L(1)\) \(\approx\) \(0.9832684317 - 0.2177624727i\)
\(L(1)\) \(\approx\) \(0.9832684317 - 0.2177624727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
71 \( 1 \)
good2 \( 1 + (-0.995 - 0.0896i)T \)
3 \( 1 + (0.473 - 0.880i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.753 - 0.657i)T \)
13 \( 1 + (0.936 - 0.351i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.983 - 0.178i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.983 + 0.178i)T \)
31 \( 1 + (0.936 - 0.351i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
41 \( 1 + (-0.963 - 0.266i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.983 - 0.178i)T \)
53 \( 1 + (-0.691 + 0.722i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (0.858 - 0.512i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (-0.393 + 0.919i)T \)
79 \( 1 + (0.936 - 0.351i)T \)
83 \( 1 + (0.936 + 0.351i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.550 - 0.834i)T \)
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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

−22.10837789567446922164670271127, −21.023015357518482777782881150408, −20.90911284938955132795607757104, −20.17488767098782659589073743291, −19.22732897472288536205132298466, −18.262929056256799547446732604103, −17.638256894676107049584016316810, −16.55360836754670700284288679912, −16.076016381012149547618327059333, −15.471898296273867216917432162708, −14.35610073222491121052035564386, −13.72632642884044369724444094987, −12.20010514741342822565185340294, −11.59412425976477113459640519230, −10.611005630086317494156212002594, −9.71747281105195635148514090640, −9.03073317994800449334695599965, −8.43437747831548517101701776232, −7.788518337786223820037280452439, −6.283095997955643928621467258549, −5.326472384575277660206177599563, −4.58253152610721018777977905595, −3.15159709109418178421066436589, −2.14747362937488377351724982677, −1.0889564462618868142741313860, 1.11198900812161079799045417810, 1.76970641979597059367150703334, 2.95722793612027152591578800665, 3.66910132835962073230874550145, 5.669347128112996060319962168, 6.51565173097553185195819561658, 7.29917355510551454381668113076, 7.9813935699744152448505866723, 8.67147817585862425910757697499, 9.8859330942637208546792233044, 10.57666060400770708763364446504, 11.43745160025801861178272163097, 12.13509760050428164231402988161, 13.52724565106072778835116285987, 13.949475537642817418795890127330, 15.040997649942883018227074712075, 15.64932294126113579939048016194, 17.17114626329367604511028698843, 17.51990182682460077703088345476, 18.3196915387394501111767796087, 18.853759840684310444163593703717, 19.70912357570758237105629563033, 20.45403604701333374278277567245, 21.11933102908091182798245752525, 22.14294055566683926672736938015

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