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Properties

Label 1-19-19.11-r0-0-0
Degree $1$
Conductor $19$
Sign $-0.305 - 0.952i$
Analytic cond. $0.0882356$
Root an. cond. $0.0882356$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(0.0882356\)
Root analytic conductor: \(0.0882356\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{19} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (0:\ ),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2653973789 - 0.3639800608i\)
\(L(\frac12)\) \(\approx\) \(0.2653973789 - 0.3639800608i\)
\(L(1)\) \(\approx\) \(0.4992035152 - 0.4022141313i\)
\(L(1)\) \(\approx\) \(0.4992035152 - 0.4022141313i\)

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

−41.1253901054614516133941262833, −39.72713785644655001036063400490, −38.042880905136483901315133727015, −37.23797040928228191804280146842, −35.182945754836544710002587601690, −34.32332760220468950085327732698, −33.353731970792871474749548817303, −32.089972383249371794155173998363, −30.2828022077364862616451162762, −28.17673136074784985593507284243, −27.2216007043267593171072508725, −26.39821209218555236472220813456, −24.577869497894815801446247298544, −23.084835327948264935378218525980, −22.003361277896269211601619608589, −19.86600550830137009502699607557, −18.04739280159696783196787643803, −16.97134418412441583103620362119, −15.255028388258225412799133289879, −14.57003800972870037766640785250, −11.396325399764076250182529561711, −10.10796142252987842080819448827, −8.1872505424540154019873653341, −6.29765489642468131575353719258, −4.40883445095759858523275964106, 1.57320808455217667944576519897, 4.639645630545534199223837216, 7.498422600723132635590660250524, 8.97228608572294362591041132672, 11.428432959283788437662983956906, 12.0908007615320473227409179149, 13.79080829985181288954638451241, 16.6734732907445739545554694964, 17.714719249995459862616200701135, 19.21004735047649087413862767539, 20.35605651057973071520226704261, 22.02787851228323272114484102839, 23.73867106697467176894410671072, 24.93890194818570857552811786818, 27.15357072471301856669666842041, 28.09764486782685251781474558821, 29.31727491172878771802277642022, 30.57284043815652044178886263982, 31.60225228856004011900831156716, 33.93294256575851837093282442897, 35.416076191824706112222999318218, 36.10464510822524108850693114442, 37.379802909321561811105896280132, 39.069391278677512493842139846169, 40.22787418304894314564986293329

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