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Properties

Label 1-817-817.64-r0-0-0
Degree $1$
Conductor $817$
Sign $0.988 + 0.154i$
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  + (−0.988 + 0.149i)2-s + (0.365 + 0.930i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + 7-s + (−0.900 + 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.623 + 0.781i)12-s + (0.826 + 0.563i)13-s + (−0.988 + 0.149i)14-s + (0.955 − 0.294i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.365 + 0.930i)3-s + (0.955 − 0.294i)4-s + (0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + 7-s + (−0.900 + 0.433i)8-s + (−0.733 + 0.680i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.623 + 0.781i)12-s + (0.826 + 0.563i)13-s + (−0.988 + 0.149i)14-s + (0.955 − 0.294i)15-s + (0.826 − 0.563i)16-s + (0.826 − 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 817 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(817\)    =    \(19 \cdot 43\)
Sign: $0.988 + 0.154i$
Analytic conductor: \(3.79413\)
Root analytic conductor: \(3.79413\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{817} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 817,\ (0:\ ),\ 0.988 + 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233910367 + 0.09583404831i\)
\(L(\frac12)\) \(\approx\) \(1.233910367 + 0.09583404831i\)
\(L(1)\) \(\approx\) \(0.9214778365 + 0.1172940945i\)
\(L(1)\) \(\approx\) \(0.9214778365 + 0.1172940945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T \)
3 \( 1 + (0.365 + 0.930i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (0.826 - 0.563i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (0.623 + 0.781i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (0.955 - 0.294i)T \)
71 \( 1 + (-0.733 - 0.680i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (0.955 + 0.294i)T \)
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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

−22.114783017196120739839081981, −21.00495486722435600186571807281, −20.44095673828483768006101670603, −19.69284936548900758221246417212, −18.640812646501469813922223194116, −18.3199441630421586476490929599, −17.70953695693985060484315557109, −17.00220249686440029956607506271, −15.62861468508070758127782378701, −14.84775592324517212265612596758, −14.317164749920824572736959601563, −13.12379356547955131887726309183, −12.17922847956097643120262443573, −11.48428915321102558535362681402, −10.575749741644077875242360172516, −9.92864559356700447405327662571, −8.62271852538487110555099007974, −7.980851822548146747111558083671, −7.38798004366829829661654013065, −6.50645393404489438692435252675, −5.67063955825390351037611261160, −3.86660662427500128891553878474, −2.76194486222538187503552451709, −2.02107223620633304594725239105, −1.11148343756346940593560393005, 0.91356835926318834134059858747, 1.961568398647381616631297883723, 3.21649653845176835462092291647, 4.36176324922526593893470944750, 5.3662073419943792656529525786, 6.0414254374011499953116032114, 7.71869910770502551015185381440, 8.283801008273167088876077013295, 8.873439676940320678343227063398, 9.675478435619693140841531303192, 10.541681202997022005377043516093, 11.45321396513133210480657923898, 11.939587819336363921714255750726, 13.66978354063967041221057111716, 14.11921765139755413854910307934, 15.33524529279797293108965042006, 15.93702238557589798597636388211, 16.606288685353875305082750502379, 17.19692849422065947747545412405, 18.196595511453084893836559445298, 19.042520985702739141807943245145, 19.96461815213648735525264247038, 20.62944290739830564086952758692, 21.247126884624420991298338645602, 21.62880788775868123292781682128

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