Properties

 Label 1-104-104.19-r0-0-0 Degree $1$ Conductor $104$ Sign $0.999 - 0.0386i$ Analytic cond. $0.482973$ Root an. cond. $0.482973$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯
 L(s)  = 1 + (−0.5 − 0.866i)3-s − i·5-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 − 0.5i)15-s + (0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s − i·21-s + (−0.5 − 0.866i)23-s − 25-s + 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$104$$    =    $$2^{3} \cdot 13$$ Sign: $0.999 - 0.0386i$ Analytic conductor: $$0.482973$$ Root analytic conductor: $$0.482973$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{104} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 104,\ (0:\ ),\ 0.999 - 0.0386i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9676930934 + 0.01868545226i$$ $$L(\frac12)$$ $$\approx$$ $$0.9676930934 + 0.01868545226i$$ $$L(1)$$ $$\approx$$ $$0.9895344955 - 0.04741307888i$$ $$L(1)$$ $$\approx$$ $$0.9895344955 - 0.04741307888i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$1$$
good3 $$1 + (-0.5 - 0.866i)T$$
5 $$1 - iT$$
7 $$1 + (0.866 + 0.5i)T$$
11 $$1 + (0.866 - 0.5i)T$$
17 $$1 + (0.5 - 0.866i)T$$
19 $$1 + (0.866 + 0.5i)T$$
23 $$1 + (-0.5 - 0.866i)T$$
29 $$1 + (0.5 + 0.866i)T$$
31 $$1 - iT$$
37 $$1 + (-0.866 + 0.5i)T$$
41 $$1 + (-0.866 + 0.5i)T$$
43 $$1 + (0.5 - 0.866i)T$$
47 $$1 - iT$$
53 $$1 - T$$
59 $$1 + (-0.866 - 0.5i)T$$
61 $$1 + (0.5 - 0.866i)T$$
67 $$1 + (-0.866 + 0.5i)T$$
71 $$1 + (-0.866 - 0.5i)T$$
73 $$1 + iT$$
79 $$1 - T$$
83 $$1 - iT$$
89 $$1 + (0.866 - 0.5i)T$$
97 $$1 + (0.866 + 0.5i)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

−29.67088626968080384642172054533, −28.36824073027195341314835986164, −27.82994708388201683550815094466, −26.97229120617451561385491599012, −25.75290764696992086175780384729, −24.404997776764095848428965546513, −23.616043850635802318867785615686, −22.44645362370606304567226385404, −21.304634779364547368921749817581, −20.560583080944362885011377532157, −19.6258184645378278088432143141, −17.585002376213150656821001283301, −17.20752264594623899232433746066, −16.0652198004158634182468935857, −14.97326066555791546669041233951, −13.78638108134130338567754359446, −12.20304890706396478193799566499, −11.40637852192964635028822365527, −10.03328497028333113217357198847, −9.057700278338838841455724834992, −7.726325132990709017264139754199, −5.91217399796355305063261290517, −4.745789265159576074864739481749, −3.88289611297301237520303759556, −1.34652327488651930984610786917, 1.59032611547197458758741245377, 3.09990239646157923027105161055, 5.15410362658805478349830946732, 6.35929671034005700082219982111, 7.38745362084218012781074582067, 8.59519333488089384551014010966, 10.383885290763299354566749952661, 11.55236638727003472255788039549, 12.13309307321626392543647890515, 13.9680920582600622766394562588, 14.40403276133497482632432403486, 16.043929047276202641003336346385, 17.34950891602712817082902158690, 18.33007020215300278533057198980, 18.849736384997758996442238657410, 20.2157200645770241377150661447, 21.7749373869791103373462180086, 22.486195200965964923592838740276, 23.55508939209076775713009567348, 24.65588424733617818771483893448, 25.32766843884309266113030475202, 26.858448672027796722145402412488, 27.653779854583120858001059371985, 28.927452391209617463888155106649, 29.86577340150726344981344796418