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Properties

Label 1-817-817.615-r0-0-0
Degree $1$
Conductor $817$
Sign $0.719 - 0.694i$
Analytic cond. $3.79413$
Root an. cond. $3.79413$
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)2-s + (0.826 + 0.563i)3-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + 6-s + (−0.5 − 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)11-s + (0.826 − 0.563i)12-s + (0.955 + 0.294i)13-s + (−0.900 − 0.433i)14-s + (−0.988 + 0.149i)15-s + (−0.733 − 0.680i)16-s + (−0.222 + 0.974i)17-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)2-s + (0.826 + 0.563i)3-s + (0.365 − 0.930i)4-s + (−0.733 + 0.680i)5-s + 6-s + (−0.5 − 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)11-s + (0.826 − 0.563i)12-s + (0.955 + 0.294i)13-s + (−0.900 − 0.433i)14-s + (−0.988 + 0.149i)15-s + (−0.733 − 0.680i)16-s + (−0.222 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(817\)    =    \(19 \cdot 43\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(3.79413\)
Root analytic conductor: \(3.79413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{817} (615, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 817,\ (0:\ ),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.634971844 - 1.063566804i\)
\(L(\frac12)\) \(\approx\) \(2.634971844 - 1.063566804i\)
\(L(1)\) \(\approx\) \(1.908957994 - 0.4594914516i\)
\(L(1)\) \(\approx\) \(1.908957994 - 0.4594914516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (0.826 + 0.563i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.955 + 0.294i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (0.365 + 0.930i)T \)
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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

−22.72562193585886081720048751895, −21.33522987465922493176900633307, −20.82578879817893851276196578108, −19.896464249909778624553322765067, −19.41255107039300152668400245881, −18.217047709698528682050069687417, −17.537646626900219726935158065863, −16.246833836094148266665905654967, −15.6638086224227350288263187846, −15.14580704675919585172202814223, −14.16743002185949785619153641841, −13.36354033968693075879239323702, −12.55035221174355835704506764702, −12.15926125976931634739537126574, −11.26417714245440899553816813730, −9.411293679503930222636883849204, −8.84454911455978622163547500898, −8.0678345304671600253739581462, −7.16120460409213370041242450706, −6.4616155575190546053449009934, −5.33687106443076896911720837564, −4.34064409374890857479014324309, −3.42273688031693975443684312077, −2.66732904951564311396176176710, −1.33065365981652813694784023370, 1.04546644378162528744657787719, 2.48895634572159004770241083005, 3.43245522328437154550847858621, 3.87519087285960914574343311708, 4.58190323453920937417814150728, 6.20479640579891004663456058000, 6.75656622413621285409570882978, 8.00097373273211687029869976200, 8.903164603734559178576119554166, 10.04039537204143682810001791116, 10.72617400054672682001680728721, 11.228329100979245481672864222566, 12.374583768667647065332716984931, 13.48087323973947381287807532042, 13.90087510010004479257731216440, 14.6843747812271386898691114402, 15.51117154001194822280021857729, 16.07360464252964678180672124500, 17.05676557966641830115334188889, 18.87803962801734606944766017377, 19.00643943685711784030995620076, 19.85334898304580119047013624076, 20.48154071285662403536920569395, 21.33494032777867041905071970336, 22.067151947645718203757216515856

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