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Properties

Label 1-647-647.607-r0-0-0
Degree $1$
Conductor $647$
Sign $-0.560 - 0.828i$
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  + (0.445 + 0.895i)2-s + (0.739 + 0.673i)3-s + (−0.602 + 0.798i)4-s + (0.0922 + 0.995i)5-s + (−0.273 + 0.961i)6-s + (−0.982 + 0.183i)7-s + (−0.982 − 0.183i)8-s + (0.0922 + 0.995i)9-s + (−0.850 + 0.526i)10-s + (0.0922 − 0.995i)11-s + (−0.982 + 0.183i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (−0.602 + 0.798i)15-s + (−0.273 − 0.961i)16-s + (0.739 + 0.673i)17-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (0.739 + 0.673i)3-s + (−0.602 + 0.798i)4-s + (0.0922 + 0.995i)5-s + (−0.273 + 0.961i)6-s + (−0.982 + 0.183i)7-s + (−0.982 − 0.183i)8-s + (0.0922 + 0.995i)9-s + (−0.850 + 0.526i)10-s + (0.0922 − 0.995i)11-s + (−0.982 + 0.183i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (−0.602 + 0.798i)15-s + (−0.273 − 0.961i)16-s + (0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $-0.560 - 0.828i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ -0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5947986925 + 1.121068525i\)
\(L(\frac12)\) \(\approx\) \(-0.5947986925 + 1.121068525i\)
\(L(1)\) \(\approx\) \(0.5878615150 + 1.043464206i\)
\(L(1)\) \(\approx\) \(0.5878615150 + 1.043464206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (0.445 + 0.895i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (0.0922 + 0.995i)T \)
7 \( 1 + (-0.982 + 0.183i)T \)
11 \( 1 + (0.0922 - 0.995i)T \)
13 \( 1 + (-0.982 + 0.183i)T \)
17 \( 1 + (0.739 + 0.673i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (-0.273 + 0.961i)T \)
29 \( 1 + (-0.273 + 0.961i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (-0.982 - 0.183i)T \)
43 \( 1 + (0.445 - 0.895i)T \)
47 \( 1 + (0.739 + 0.673i)T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (-0.850 + 0.526i)T \)
61 \( 1 + (0.739 - 0.673i)T \)
67 \( 1 + (0.739 - 0.673i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (-0.850 - 0.526i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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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

−22.459283253612848660531830739699, −21.165749681064380394537624954449, −20.60036172190533423941561120744, −19.90196702686286760143371264125, −19.39333799638087801204795281588, −18.53041247434870035831164900091, −17.59615556635951597874061881169, −16.61228189385400912750769416021, −15.4580864031144862071513195269, −14.50331753551387510930269930566, −13.831566273402886356293028120676, −12.78260272738866949279065925972, −12.49229202711970386661043530218, −11.908928961335222631071122913734, −10.07136450443250172463830951371, −9.765991638574386001736253194040, −8.87195955702291786918770871842, −7.81376090273606431609729190046, −6.72400810621358672328383287414, −5.62607728548784593692070038207, −4.53576631000969519048503764509, −3.63219380091252939811907067538, −2.55809497671590366610485551688, −1.715150121042588167058721398975, −0.46355441322943065652241124432, 2.50002664432941549746957804677, 3.27174466254366987524012757468, 3.88073765842690308765186015356, 5.22592050429262274850997043289, 6.09504975614280688817384666413, 7.052110539014863010172149588849, 7.83940302548037349490960130894, 8.932208818341891955617274604606, 9.63035000918000476272295811807, 10.51820278250615281134586464464, 11.70541790805293603046273994631, 12.89333080683664392603965218927, 13.75543570418507914008690169096, 14.289554823526049412135068585889, 15.27538635204560584758352925730, 15.584405226502409138001386376463, 16.6982438816601797688100762512, 17.25533636999307150359505757269, 18.80373685593825637394526463728, 19.03917787415578143821268994034, 20.122785943007847858516585662836, 21.485619693106299611578472125938, 21.96705982115050073703930826094, 22.24588143286600631765308301909, 23.45237329781869607621907931888

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