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Properties

Label 1-373-373.21-r0-0-0
Degree $1$
Conductor $373$
Sign $0.118 + 0.992i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.470 + 0.882i)2-s + (0.943 − 0.331i)3-s + (−0.557 + 0.830i)4-s + (0.890 + 0.455i)5-s + (0.736 + 0.676i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (0.780 − 0.625i)9-s + (0.0168 + 0.999i)10-s + (0.585 + 0.810i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.0843 + 0.996i)14-s + (0.990 + 0.134i)15-s + (−0.378 − 0.925i)16-s + (−0.440 − 0.897i)17-s + ⋯
L(s)  = 1  + (0.470 + 0.882i)2-s + (0.943 − 0.331i)3-s + (−0.557 + 0.830i)4-s + (0.890 + 0.455i)5-s + (0.736 + 0.676i)6-s + (0.918 + 0.394i)7-s + (−0.994 − 0.101i)8-s + (0.780 − 0.625i)9-s + (0.0168 + 0.999i)10-s + (0.585 + 0.810i)11-s + (−0.250 + 0.968i)12-s + (−0.994 + 0.101i)13-s + (0.0843 + 0.996i)14-s + (0.990 + 0.134i)15-s + (−0.378 − 0.925i)16-s + (−0.440 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.118 + 0.992i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.118 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.916122935 + 1.701470966i\)
\(L(\frac12)\) \(\approx\) \(1.916122935 + 1.701470966i\)
\(L(1)\) \(\approx\) \(1.702661413 + 0.9601287115i\)
\(L(1)\) \(\approx\) \(1.702661413 + 0.9601287115i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.470 + 0.882i)T \)
3 \( 1 + (0.943 - 0.331i)T \)
5 \( 1 + (0.890 + 0.455i)T \)
7 \( 1 + (0.918 + 0.394i)T \)
11 \( 1 + (0.585 + 0.810i)T \)
13 \( 1 + (-0.994 + 0.101i)T \)
17 \( 1 + (-0.440 - 0.897i)T \)
19 \( 1 + (-0.612 - 0.790i)T \)
23 \( 1 + (-0.758 + 0.651i)T \)
29 \( 1 + (0.638 - 0.769i)T \)
31 \( 1 + (-0.250 - 0.968i)T \)
37 \( 1 + (-0.315 - 0.948i)T \)
41 \( 1 + (-0.954 + 0.299i)T \)
43 \( 1 + (-0.557 + 0.830i)T \)
47 \( 1 + (-0.905 + 0.425i)T \)
53 \( 1 + (0.780 + 0.625i)T \)
59 \( 1 + (-0.985 - 0.168i)T \)
61 \( 1 + (0.943 - 0.331i)T \)
67 \( 1 + (-0.0506 - 0.998i)T \)
71 \( 1 + (0.997 - 0.0675i)T \)
73 \( 1 + (-0.931 + 0.363i)T \)
79 \( 1 + (0.0168 + 0.999i)T \)
83 \( 1 + (0.780 + 0.625i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.440 - 0.897i)T \)
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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

−24.38710876354367844871124472825, −23.72080493837089080685513145560, −22.02461647841223827569452829705, −21.75416704120690616818158851429, −20.91614742712662876306414730438, −20.16989574545065842923131198284, −19.52157615448383840370134446951, −18.47464823673571992225690853603, −17.44336718047186579634203031299, −16.502502936906653953194758047724, −14.95863384387146712989588012773, −14.3606852363717928555094203540, −13.77817415850671286266208012327, −12.84254796661856271895011519274, −11.88510622112921972926824947527, −10.392997758650436483145356286839, −10.21254677536882821009590416294, −8.76377671487752790417354377076, −8.410623987226530859605223192229, −6.59542360266090134473489195205, −5.25761688242544757752477181338, −4.46275227863912364271176083216, −3.45616350913303561144953173945, −2.13268697870762790980237883974, −1.47354103231897319824848832812, 1.96932581740152191259373897607, 2.69773095075277032176965777264, 4.22707802747625932444480179055, 5.10193643574672245285601919939, 6.44026143295001672310345634463, 7.18644198021271385358115684441, 8.06108005713813647300671802801, 9.218183073766872771633212513544, 9.73239358157953405212263768648, 11.55069460137322718270374498293, 12.51608238542385720660591786844, 13.55028087607695006374735046003, 14.18528248238305959437526514153, 14.90703556350955146076669570099, 15.450348399112179318363965178642, 17.0330089540829473970184783050, 17.78328462529074826011696952724, 18.2771667310260967777500609917, 19.57177006410134249415877205433, 20.64161485805213774153857712366, 21.52739459024296278722694220115, 22.06405523529672763212781717079, 23.203650138849119065185191748497, 24.39827849237048871088180179015, 24.73494259128445485530045364870

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