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Properties

Label 1-373-373.11-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.999 + 0.0288i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (−0.543 − 0.839i)2-s + (−0.0168 − 0.999i)3-s + (−0.409 + 0.912i)4-s + (−0.331 + 0.943i)5-s + (−0.830 + 0.557i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (−0.999 + 0.0337i)9-s + (0.972 − 0.234i)10-s + (−0.625 + 0.780i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (0.925 + 0.378i)14-s + (0.948 + 0.315i)15-s + (−0.664 − 0.747i)16-s + (0.994 − 0.101i)17-s + ⋯
L(s)  = 1  + (−0.543 − 0.839i)2-s + (−0.0168 − 0.999i)3-s + (−0.409 + 0.912i)4-s + (−0.331 + 0.943i)5-s + (−0.830 + 0.557i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (−0.999 + 0.0337i)9-s + (0.972 − 0.234i)10-s + (−0.625 + 0.780i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (0.925 + 0.378i)14-s + (0.948 + 0.315i)15-s + (−0.664 − 0.747i)16-s + (0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.999 + 0.0288i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.999 + 0.0288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001290041494 - 0.08942810690i\)
\(L(\frac12)\) \(\approx\) \(0.001290041494 - 0.08942810690i\)
\(L(1)\) \(\approx\) \(0.5261006791 - 0.1533917432i\)
\(L(1)\) \(\approx\) \(0.5261006791 - 0.1533917432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.543 - 0.839i)T \)
3 \( 1 + (-0.0168 - 0.999i)T \)
5 \( 1 + (-0.331 + 0.943i)T \)
7 \( 1 + (-0.820 + 0.571i)T \)
11 \( 1 + (-0.625 + 0.780i)T \)
13 \( 1 + (-0.151 + 0.988i)T \)
17 \( 1 + (0.994 - 0.101i)T \)
19 \( 1 + (0.201 - 0.979i)T \)
23 \( 1 + (0.485 + 0.874i)T \)
29 \( 1 + (0.963 + 0.266i)T \)
31 \( 1 + (-0.918 + 0.394i)T \)
37 \( 1 + (-0.217 + 0.975i)T \)
41 \( 1 + (-0.440 + 0.897i)T \)
43 \( 1 + (-0.912 - 0.409i)T \)
47 \( 1 + (0.134 - 0.990i)T \)
53 \( 1 + (0.0337 - 0.999i)T \)
59 \( 1 + (0.713 - 0.701i)T \)
61 \( 1 + (-0.999 + 0.0168i)T \)
67 \( 1 + (-0.651 - 0.758i)T \)
71 \( 1 + (-0.585 + 0.810i)T \)
73 \( 1 + (0.470 + 0.882i)T \)
79 \( 1 + (-0.234 - 0.972i)T \)
83 \( 1 + (-0.999 - 0.0337i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.101 - 0.994i)T \)
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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

−25.08915740383369402832731723754, −23.96105169208527029246520321005, −23.10714998199026066350658602231, −22.57809046135844179005521995907, −21.18795476855362788073751811821, −20.36592560562523832210992790473, −19.615842146706713386786505258430, −18.68148193284022242458927377384, −17.35926204687778946981062174962, −16.52597497829141639820705120468, −16.2416061183987557600208279849, −15.39267817181097521218057654361, −14.37589112664490914620531755225, −13.37228523695965923285268006788, −12.3080612733965905286442050450, −10.71334455850696539067267474063, −10.18881696588148959688265182302, −9.23286632332385392397451290384, −8.326914590733616035400545776558, −7.587990485159049811559355606645, −5.95000861060582349576323621548, −5.3775984220084166975060012125, −4.24621304593807277006743099013, −3.17173659809297873152115873398, −0.85283017649068016901803469401, 0.04075334128270818548985181274, 1.67692967206265593937863143658, 2.71595636312831760819349318100, 3.37574872239161128182329746990, 5.11761504988160787838481081539, 6.74552378146524675429931253861, 7.204931201683007307046347915181, 8.30801042189108236068314175721, 9.41504249122550596996105927743, 10.289255520386146664108127127869, 11.52708511511153947521769925675, 11.9929253365900938057076588504, 12.971533164848988905668526159616, 13.77445225396259336973422414580, 14.9407713062316649406808804349, 16.127695497431711807151887700360, 17.23418100049630167529012012388, 18.24300633099665063495401412409, 18.63858726471048859095572316156, 19.43745582980464391409650651842, 20.0166309997814211898237418296, 21.41196410653399088719845271569, 22.12192147409302117913634909321, 23.12205263501107863362584761265, 23.61216782595037546024881547105

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