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Properties

Label 1-373-373.53-r1-0-0
Degree $1$
Conductor $373$
Sign $0.928 - 0.370i$
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  + (0.982 − 0.184i)2-s + (0.972 − 0.234i)3-s + (0.931 − 0.363i)4-s + (0.999 + 0.0168i)5-s + (0.912 − 0.409i)6-s + (0.612 + 0.790i)7-s + (0.848 − 0.528i)8-s + (0.890 − 0.455i)9-s + (0.985 − 0.168i)10-s + (0.0337 − 0.999i)11-s + (0.820 − 0.571i)12-s + (−0.528 + 0.848i)13-s + (0.747 + 0.664i)14-s + (0.975 − 0.217i)15-s + (0.736 − 0.676i)16-s + (−0.151 + 0.988i)17-s + ⋯
L(s)  = 1  + (0.982 − 0.184i)2-s + (0.972 − 0.234i)3-s + (0.931 − 0.363i)4-s + (0.999 + 0.0168i)5-s + (0.912 − 0.409i)6-s + (0.612 + 0.790i)7-s + (0.848 − 0.528i)8-s + (0.890 − 0.455i)9-s + (0.985 − 0.168i)10-s + (0.0337 − 0.999i)11-s + (0.820 − 0.571i)12-s + (−0.528 + 0.848i)13-s + (0.747 + 0.664i)14-s + (0.975 − 0.217i)15-s + (0.736 − 0.676i)16-s + (−0.151 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.928 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(7.035791040 - 1.350225730i\)
\(L(\frac12)\) \(\approx\) \(7.035791040 - 1.350225730i\)
\(L(1)\) \(\approx\) \(3.294320994 - 0.4824513625i\)
\(L(1)\) \(\approx\) \(3.294320994 - 0.4824513625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.982 - 0.184i)T \)
3 \( 1 + (0.972 - 0.234i)T \)
5 \( 1 + (0.999 + 0.0168i)T \)
7 \( 1 + (0.612 + 0.790i)T \)
11 \( 1 + (0.0337 - 0.999i)T \)
13 \( 1 + (-0.528 + 0.848i)T \)
17 \( 1 + (-0.151 + 0.988i)T \)
19 \( 1 + (0.299 + 0.954i)T \)
23 \( 1 + (-0.724 - 0.688i)T \)
29 \( 1 + (-0.801 - 0.598i)T \)
31 \( 1 + (-0.820 - 0.571i)T \)
37 \( 1 + (-0.997 + 0.0675i)T \)
41 \( 1 + (-0.994 - 0.101i)T \)
43 \( 1 + (0.363 + 0.931i)T \)
47 \( 1 + (0.948 + 0.315i)T \)
53 \( 1 + (-0.455 + 0.890i)T \)
59 \( 1 + (0.117 - 0.993i)T \)
61 \( 1 + (-0.234 - 0.972i)T \)
67 \( 1 + (-0.485 - 0.874i)T \)
71 \( 1 + (-0.780 + 0.625i)T \)
73 \( 1 + (-0.839 + 0.543i)T \)
79 \( 1 + (-0.168 - 0.985i)T \)
83 \( 1 + (0.890 + 0.455i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.988 + 0.151i)T \)
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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

−24.46264358993621823538842892329, −23.762590691843669388171551333469, −22.42129977607629906479393235230, −21.93721627809282653654572240682, −20.7755606235252732028893620790, −20.40291376400275405757757337905, −19.745240978014382604472268548360, −18.03498963188525132178558626498, −17.368857061963312961622071156517, −16.2245738032821071052461948802, −15.19047045253775599070770685270, −14.53375005809133151916591845199, −13.691897780285599366139346334421, −13.22334172968566072388203724141, −12.09772208995953296580242451639, −10.69176931791672839255962335985, −9.99279095877854187531582088786, −8.85829229738145744515759820762, −7.34321112881445904351724066353, −7.17344901614230204325510666944, −5.34144689221341369635974458219, −4.762048609333888121706229802894, −3.549275007188867796659786494106, −2.44538628798633854604625743626, −1.58088467465010395917118724202, 1.65333021772493734622692417869, 2.11679074415839001574586524389, 3.28454930769726512079657804330, 4.413257506050753302091881072957, 5.701881478402306828998960540, 6.35196216394472509033952350383, 7.702946517434510480071846529583, 8.74680091448043928543628585455, 9.73869104272730581282433438946, 10.80811132388650449063938798548, 12.00606558703672225166225339098, 12.79592129744604021790131698587, 13.793379263303078549570916092271, 14.34253911875517576068720673200, 14.96346668221462033731998558193, 16.12348412029286574238661879969, 17.17213609717765131440256082055, 18.66364508787244309141679719323, 18.96458393579499526389168842053, 20.31640965764743692559494346360, 20.963685945239957462307361018912, 21.743238541917238106174384958031, 22.12538139767570860753811173363, 23.82204272109027873166714408817, 24.435570690796218769693212017003

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