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Properties

Label 1-647-647.316-r0-0-0
Degree $1$
Conductor $647$
Sign $-0.496 - 0.868i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (0.739 + 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 + 0.995i)4-s + (0.445 − 0.895i)5-s + (−0.982 − 0.183i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s + (0.932 − 0.361i)10-s + (0.445 + 0.895i)11-s + (−0.602 − 0.798i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.0922 + 0.995i)15-s + (−0.982 + 0.183i)16-s + (−0.850 + 0.526i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (−0.850 + 0.526i)3-s + (0.0922 + 0.995i)4-s + (0.445 − 0.895i)5-s + (−0.982 − 0.183i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.445 − 0.895i)9-s + (0.932 − 0.361i)10-s + (0.445 + 0.895i)11-s + (−0.602 − 0.798i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.0922 + 0.995i)15-s + (−0.982 + 0.183i)16-s + (−0.850 + 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $-0.496 - 0.868i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ -0.496 - 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01232743872 + 0.02125264406i\)
\(L(\frac12)\) \(\approx\) \(0.01232743872 + 0.02125264406i\)
\(L(1)\) \(\approx\) \(0.7774378044 + 0.3176477203i\)
\(L(1)\) \(\approx\) \(0.7774378044 + 0.3176477203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (0.739 + 0.673i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (0.445 - 0.895i)T \)
7 \( 1 + (-0.602 - 0.798i)T \)
11 \( 1 + (0.445 + 0.895i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (-0.850 + 0.526i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (-0.982 - 0.183i)T \)
29 \( 1 + (-0.982 - 0.183i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (-0.602 + 0.798i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (-0.850 + 0.526i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (0.932 - 0.361i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (-0.850 - 0.526i)T \)
71 \( 1 + (-0.982 - 0.183i)T \)
73 \( 1 + (0.739 + 0.673i)T \)
79 \( 1 + (0.739 - 0.673i)T \)
83 \( 1 + (0.932 + 0.361i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.932 - 0.361i)T \)
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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

−22.75083537584408651143551325723, −22.25532602558975399792182414501, −21.87909970109611727192869991207, −21.07135121472057309396746526585, −19.530255592668722222479695797823, −19.088245177696697082035674558006, −18.49250883989754882914085977677, −17.589576738058736809570080174302, −16.499807850184306980369380722490, −15.57120849018268021219880780181, −14.63605344940561676302984096266, −13.70401006857891411038188284329, −13.14587667968950515447577397950, −12.04255902270101571120329471823, −11.53345595386433178701824114143, −10.76201719622323998945612361861, −9.83634303110297352077959203790, −8.93567501195230405805979875395, −7.18409179812960445121119773390, −6.29812905517480099223211722094, −6.00504438423015553538623390399, −4.859966061754377988219558739485, −3.66728657073329300930733746875, −2.44211601949210861551887989212, −1.87810210021740697263502991059, 0.00943881260756531898873439721, 1.89003925955589335724801612074, 3.58936121271901626901153462489, 4.4108318615201386420930949897, 4.99138823114260284611806504173, 6.11373738379390408282560273588, 6.655141622349790305805852398268, 7.78843235316733325107994243647, 8.97169873430613532617047311491, 9.89703823636396175917704771978, 10.70534626741414462287362428501, 12.04321481975725946838054332188, 12.63333714127425394485524443047, 13.19829527008550507529881490730, 14.365052381546442708382134165560, 15.28557717722688652905726949722, 16.00447287398232914715242350305, 16.83251459039422762284096108092, 17.362994187959246687807256638431, 17.806284361541320082315678598, 19.66896043568234774243488675641, 20.42734141172859167936044798611, 21.09301110453224846288895728191, 22.185519526646284040185543925478, 22.47075343627020411397934650516

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