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Properties

Label 1-104-104.11-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 $0$

Origins

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Normalization:  

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: $\chi_{104} (11, \cdot )$
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

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

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