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Properties

Label 1-19-19.12-r1-0-0
Degree $1$
Conductor $19$
Sign $-0.0977 - 0.995i$
Analytic cond. $2.04183$
Root an. cond. $2.04183$
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)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(2.04183\)
Root analytic conductor: \(2.04183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (1:\ ),\ -0.0977 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141713410 - 1.259329153i\)
\(L(\frac12)\) \(\approx\) \(1.141713410 - 1.259329153i\)
\(L(1)\) \(\approx\) \(1.171924367 - 0.8392882274i\)
\(L(1)\) \(\approx\) \(1.171924367 - 0.8392882274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−40.33680660779378355267225111743, −39.76894410016231464164242638487, −38.05242432076441457165335384665, −36.5022579806659351786055901255, −35.16236977504697205545570171702, −33.62753357503607261543864393779, −32.62001292822573068813174805871, −31.578698570217295819451866599956, −30.47704666509995754693122282020, −27.72092988376729261011420977864, −27.120891469393447115918289335444, −25.33493562120286699599703462346, −24.33675295044782344412008447845, −22.719775693147160589467185167057, −21.23363307428051330851947110848, −20.09849341548238658402274720182, −17.51583683932556909175314403764, −16.13947603054479514636171405548, −15.03416467770451859391945579129, −13.65167300143404232055779003349, −11.652112305640210271450649631705, −9.05044664237128583139486041143, −7.91237680847372134581151384136, −5.21098682123913118417296988140, −3.92233777169271607630467815507, 1.84482028253497795644153396587, 3.85721465144342870153484458138, 6.59057174491198024235851891531, 8.66158334459240346469061609330, 11.01487083332929131354584537556, 12.15039477144890800518029889077, 14.02874793732491892878769755135, 14.78188146424847670694236087619, 17.93416506123136272890579987888, 19.041512272109271623418469717860, 20.1747867126173240804468052608, 21.80462104465686668374337708320, 23.37101286354787981780300594449, 24.38992397140648675024543132618, 26.39745884254095791544132708510, 27.8742604313350682280978621468, 29.598036641016633248653425019095, 30.675111054504296956347579188758, 31.12432491388082169305469353114, 33.00556900783129098082609509108, 34.73597277755550748881356626955, 36.26613511185089851317247983395, 37.52787904583933348724357455117, 38.3557777215390223940140068752, 40.156433047727811516187470182755

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