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Properties

Label 1-373-373.194-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.237 + 0.971i$
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  + (0.839 + 0.543i)2-s + (0.0168 − 0.999i)3-s + (0.409 + 0.912i)4-s + (−0.943 + 0.331i)5-s + (0.557 − 0.830i)6-s + (0.820 + 0.571i)7-s + (−0.151 + 0.988i)8-s + (−0.999 − 0.0337i)9-s + (−0.972 − 0.234i)10-s + (−0.780 + 0.625i)11-s + (0.918 − 0.394i)12-s + (0.151 + 0.988i)13-s + (0.378 + 0.925i)14-s + (0.315 + 0.948i)15-s + (−0.664 + 0.747i)16-s + (−0.994 − 0.101i)17-s + ⋯
L(s)  = 1  + (0.839 + 0.543i)2-s + (0.0168 − 0.999i)3-s + (0.409 + 0.912i)4-s + (−0.943 + 0.331i)5-s + (0.557 − 0.830i)6-s + (0.820 + 0.571i)7-s + (−0.151 + 0.988i)8-s + (−0.999 − 0.0337i)9-s + (−0.972 − 0.234i)10-s + (−0.780 + 0.625i)11-s + (0.918 − 0.394i)12-s + (0.151 + 0.988i)13-s + (0.378 + 0.925i)14-s + (0.315 + 0.948i)15-s + (−0.664 + 0.747i)16-s + (−0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.237 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9248206511 + 1.178597571i\)
\(L(\frac12)\) \(\approx\) \(0.9248206511 + 1.178597571i\)
\(L(1)\) \(\approx\) \(1.232948885 + 0.5207562136i\)
\(L(1)\) \(\approx\) \(1.232948885 + 0.5207562136i\)

Euler product

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

−24.11239944867391394785225592234, −23.16152328270685442374285389018, −22.82269396061365465701023953004, −21.37889548790264336206759347136, −21.14099418567940930101288200159, −20.00869016529068740692510929766, −19.73509749278109505903983971221, −18.329365601504121411214766309869, −17.06012259438300046395121156806, −16.029701029557718207670955788295, −15.30408818773014756019541184364, −14.73076622011648561929763753346, −13.54985119212067213139678815695, −12.711184296186078031888312348908, −11.39277536206637900381302814439, −10.94378563488856160504051286150, −10.25835727373541688060670708099, −8.72923303131361911100943982304, −7.96160784408720415333665691234, −6.40316107993390494911923911445, −5.00508519318800147293345053395, −4.61885888652414531603200876135, −3.5875684278538203253003545882, −2.61333852677127041604859268967, −0.66620245860303628900864466812, 1.962971093004637866165457136951, 2.83415000437441035550108507307, 4.3062332856832171169085769492, 5.13645203557273705607698365305, 6.53926367393562092437703304563, 7.07929406353774190500786793482, 8.17305667536920774349054435953, 8.63738483459022690875578440955, 10.88099558785172700755312959046, 11.684529760089781770138922038051, 12.25690792800598987686480182398, 13.26321587256088638026138062310, 14.14384695955113754244944685415, 15.10641421321113016217135746783, 15.58523092618394085413261677162, 16.9349633605433238134518008866, 17.7884258563081736631977127277, 18.64818033264436689577625808783, 19.551617288873444295134347558751, 20.62991245520140290802843464242, 21.45341028398892503631407362820, 22.58454853535205381446129048700, 23.4427020928991335713251550922, 23.78970744835444227099995948594, 24.64257868905603899983993779187

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