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Properties

Label 1-373-373.157-r1-0-0
Degree $1$
Conductor $373$
Sign $0.334 + 0.942i$
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  + (−0.897 + 0.440i)2-s + (−0.151 − 0.988i)3-s + (0.612 − 0.790i)4-s + (0.101 + 0.994i)5-s + (0.571 + 0.820i)6-s + (−0.688 − 0.724i)7-s + (−0.201 + 0.979i)8-s + (−0.954 + 0.299i)9-s + (−0.528 − 0.848i)10-s + (−0.201 − 0.979i)11-s + (−0.874 − 0.485i)12-s + (−0.979 + 0.201i)13-s + (0.937 + 0.347i)14-s + (0.968 − 0.250i)15-s + (−0.250 − 0.968i)16-s + (0.612 − 0.790i)17-s + ⋯
L(s)  = 1  + (−0.897 + 0.440i)2-s + (−0.151 − 0.988i)3-s + (0.612 − 0.790i)4-s + (0.101 + 0.994i)5-s + (0.571 + 0.820i)6-s + (−0.688 − 0.724i)7-s + (−0.201 + 0.979i)8-s + (−0.954 + 0.299i)9-s + (−0.528 − 0.848i)10-s + (−0.201 − 0.979i)11-s + (−0.874 − 0.485i)12-s + (−0.979 + 0.201i)13-s + (0.937 + 0.347i)14-s + (0.968 − 0.250i)15-s + (−0.250 − 0.968i)16-s + (0.612 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.334 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3100107678 + 0.2189231383i\)
\(L(\frac12)\) \(\approx\) \(0.3100107678 + 0.2189231383i\)
\(L(1)\) \(\approx\) \(0.5174452288 - 0.05454646432i\)
\(L(1)\) \(\approx\) \(0.5174452288 - 0.05454646432i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.897 + 0.440i)T \)
3 \( 1 + (-0.151 - 0.988i)T \)
5 \( 1 + (0.101 + 0.994i)T \)
7 \( 1 + (-0.688 - 0.724i)T \)
11 \( 1 + (-0.201 - 0.979i)T \)
13 \( 1 + (-0.979 + 0.201i)T \)
17 \( 1 + (0.612 - 0.790i)T \)
19 \( 1 + (-0.968 - 0.250i)T \)
23 \( 1 + (0.988 + 0.151i)T \)
29 \( 1 + (-0.758 + 0.651i)T \)
31 \( 1 + (0.874 - 0.485i)T \)
37 \( 1 + (-0.918 - 0.394i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (-0.790 - 0.612i)T \)
47 \( 1 + (-0.937 + 0.347i)T \)
53 \( 1 + (-0.299 + 0.954i)T \)
59 \( 1 + (0.758 - 0.651i)T \)
61 \( 1 + (0.988 - 0.151i)T \)
67 \( 1 + (0.101 + 0.994i)T \)
71 \( 1 + (0.612 + 0.790i)T \)
73 \( 1 + (-0.954 - 0.299i)T \)
79 \( 1 + (0.848 - 0.528i)T \)
83 \( 1 + (-0.954 - 0.299i)T \)
89 \( 1 - T \)
97 \( 1 + (0.790 + 0.612i)T \)
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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

−24.58741250236357190900540835743, −23.12948400355605496464620829536, −22.24652840427989014422536533851, −21.129292291678423035436410160713, −20.983136696135255555284498336677, −19.70824803251306332582967117912, −19.286297822091641080265289278626, −17.83162270926536467520783376748, −17.01925871996291206193277089410, −16.545956880702158113367789607179, −15.45664667505981483181847139311, −14.90804344426220993019871895407, −12.89391034540681736034183941467, −12.434266700072561880872081097918, −11.499314919017040161724227730445, −10.06376612744176762629164655550, −9.8359548563361901515408498305, −8.83045093636954109859917355349, −8.07671306208869385557700035918, −6.56809804257539761412780618306, −5.34151744271076815545550359454, −4.31351777249907329288646972394, −3.067540376324478788265047606199, −1.91765356870130130163212124795, −0.20104830063367724461335572492, 0.76805913700049904818512335500, 2.29010447824567650642265276156, 3.19238659964720418123376570815, 5.34502691225902129462145128805, 6.393166980671778429102095137214, 7.057601944882646736277609041259, 7.65755386628934584148961365900, 8.89267400914447894919933405131, 10.01623756265072828850800456945, 10.88036780303805218993742574943, 11.6356909474472192971207775500, 12.99703874641217663058014521700, 14.0199389199219136006057913152, 14.63197390538159443895311522043, 15.90832517936948723602766829781, 16.94467789322132431116581916104, 17.406292279512832425465856896472, 18.61173639978362141341810310236, 19.08419203263278516696506305898, 19.58587165131884652319424231740, 20.84608779201539734898307525389, 22.239296171333050639776713661005, 23.11386012376305556863865374553, 23.76204564628769431684348083052, 24.705462388452768103793413178599

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