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Properties

Label 1-163-163.26-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.524 + 0.851i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (−0.431 + 0.902i)2-s + (−0.875 + 0.483i)3-s + (−0.627 − 0.778i)4-s + (0.396 − 0.918i)5-s + (−0.0581 − 0.998i)6-s + (−0.910 + 0.413i)7-s + (0.973 − 0.230i)8-s + (0.533 − 0.845i)9-s + (0.657 + 0.753i)10-s + (0.856 + 0.516i)11-s + (0.925 + 0.378i)12-s + (−0.686 + 0.727i)13-s + (0.0193 − 0.999i)14-s + (0.0968 + 0.995i)15-s + (−0.211 + 0.977i)16-s + (−0.286 + 0.957i)17-s + ⋯
L(s)  = 1  + (−0.431 + 0.902i)2-s + (−0.875 + 0.483i)3-s + (−0.627 − 0.778i)4-s + (0.396 − 0.918i)5-s + (−0.0581 − 0.998i)6-s + (−0.910 + 0.413i)7-s + (0.973 − 0.230i)8-s + (0.533 − 0.845i)9-s + (0.657 + 0.753i)10-s + (0.856 + 0.516i)11-s + (0.925 + 0.378i)12-s + (−0.686 + 0.727i)13-s + (0.0193 − 0.999i)14-s + (0.0968 + 0.995i)15-s + (−0.211 + 0.977i)16-s + (−0.286 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.524 + 0.851i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2686713321 + 0.4808989385i\)
\(L(\frac12)\) \(\approx\) \(0.2686713321 + 0.4808989385i\)
\(L(1)\) \(\approx\) \(0.5132633416 + 0.3329338529i\)
\(L(1)\) \(\approx\) \(0.5132633416 + 0.3329338529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.431 + 0.902i)T \)
3 \( 1 + (-0.875 + 0.483i)T \)
5 \( 1 + (0.396 - 0.918i)T \)
7 \( 1 + (-0.910 + 0.413i)T \)
11 \( 1 + (0.856 + 0.516i)T \)
13 \( 1 + (-0.686 + 0.727i)T \)
17 \( 1 + (-0.286 + 0.957i)T \)
19 \( 1 + (0.713 + 0.700i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.565 + 0.824i)T \)
31 \( 1 + (0.597 + 0.802i)T \)
37 \( 1 + (-0.835 - 0.549i)T \)
41 \( 1 + (-0.627 + 0.778i)T \)
43 \( 1 + (0.323 + 0.946i)T \)
47 \( 1 + (0.996 - 0.0774i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.973 + 0.230i)T \)
67 \( 1 + (0.987 + 0.154i)T \)
71 \( 1 + (-0.981 - 0.192i)T \)
73 \( 1 + (0.987 - 0.154i)T \)
79 \( 1 + (-0.999 + 0.0387i)T \)
83 \( 1 + (0.813 + 0.581i)T \)
89 \( 1 + (0.856 - 0.516i)T \)
97 \( 1 + (0.466 - 0.884i)T \)
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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

−27.40503459600411410865008966788, −26.77576041120494204481622672317, −25.6300941505753327008924065874, −24.60115522880839392096527881561, −23.00915425432967708920772870906, −22.34439702317993427622038977495, −21.97973316456239729403389201805, −20.40000103160659749281420341553, −19.22071689902135896790427593630, −18.78050842851451948431642555794, −17.46367556172951704493794388243, −17.1220009308476460918595498687, −15.69698503891444491085110813291, −13.82351555208749160951583071464, −13.2604542535802905318914992248, −11.94511090704464241158607911112, −11.19399069108781104782731147228, −10.1724335773024815684113897131, −9.36381710388823561755214459835, −7.47661009376566561180353103657, −6.73725797542324358647553274810, −5.29316713787981473803325063423, −3.57201600848827650272088364911, −2.41867087227956964248337850605, −0.66103258737372779513241381461, 1.383537332254514186916520678172, 4.102527031619368725849767187645, 5.13287353750815124081075561738, 6.13787443322033891031215369310, 6.991669035530956728581179158535, 8.84408540103708299525638945285, 9.46686682068431102826418914020, 10.35798167022098318302772037692, 12.066426607776673605001752458894, 12.85228939542036761770756093211, 14.36015383143516851170099488610, 15.4433171182324200704292817338, 16.482174195581299841553300534206, 16.88508186461656997280298905625, 17.81666221960776362690596664650, 19.03800641509727374619029963355, 20.09759714258986923602351746973, 21.57515353834389080503914767303, 22.37872716314146271290037373556, 23.26949205344431549291317866176, 24.37617089145497567179688587178, 25.03179481380771405202365167564, 26.17959343124167511034718510476, 27.09194970211419103781309753455, 28.18402870888448656750693338538

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