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Properties

Label 1-373-373.134-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.995 + 0.0943i$
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  + (0.713 + 0.701i)2-s + (−0.905 + 0.425i)3-s + (0.0168 + 0.999i)4-s + (0.801 − 0.598i)5-s + (−0.943 − 0.331i)6-s + (−0.994 + 0.101i)7-s + (−0.688 + 0.724i)8-s + (0.638 − 0.769i)9-s + (0.990 + 0.134i)10-s + (−0.283 + 0.959i)11-s + (−0.440 − 0.897i)12-s + (0.688 + 0.724i)13-s + (−0.780 − 0.625i)14-s + (−0.470 + 0.882i)15-s + (−0.999 + 0.0337i)16-s + (−0.874 − 0.485i)17-s + ⋯
L(s)  = 1  + (0.713 + 0.701i)2-s + (−0.905 + 0.425i)3-s + (0.0168 + 0.999i)4-s + (0.801 − 0.598i)5-s + (−0.943 − 0.331i)6-s + (−0.994 + 0.101i)7-s + (−0.688 + 0.724i)8-s + (0.638 − 0.769i)9-s + (0.990 + 0.134i)10-s + (−0.283 + 0.959i)11-s + (−0.440 − 0.897i)12-s + (0.688 + 0.724i)13-s + (−0.780 − 0.625i)14-s + (−0.470 + 0.882i)15-s + (−0.999 + 0.0337i)16-s + (−0.874 − 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.995 + 0.0943i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.995 + 0.0943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04716751063 + 0.9977045512i\)
\(L(\frac12)\) \(\approx\) \(0.04716751063 + 0.9977045512i\)
\(L(1)\) \(\approx\) \(0.7566704739 + 0.6788169310i\)
\(L(1)\) \(\approx\) \(0.7566704739 + 0.6788169310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.713 + 0.701i)T \)
3 \( 1 + (-0.905 + 0.425i)T \)
5 \( 1 + (0.801 - 0.598i)T \)
7 \( 1 + (-0.994 + 0.101i)T \)
11 \( 1 + (-0.283 + 0.959i)T \)
13 \( 1 + (0.688 + 0.724i)T \)
17 \( 1 + (-0.874 - 0.485i)T \)
19 \( 1 + (-0.528 + 0.848i)T \)
23 \( 1 + (-0.820 + 0.571i)T \)
29 \( 1 + (0.736 + 0.676i)T \)
31 \( 1 + (-0.440 + 0.897i)T \)
37 \( 1 + (-0.839 + 0.543i)T \)
41 \( 1 + (-0.758 + 0.651i)T \)
43 \( 1 + (-0.0168 - 0.999i)T \)
47 \( 1 + (0.931 + 0.363i)T \)
53 \( 1 + (-0.638 - 0.769i)T \)
59 \( 1 + (0.217 - 0.975i)T \)
61 \( 1 + (0.905 - 0.425i)T \)
67 \( 1 + (-0.918 - 0.394i)T \)
71 \( 1 + (0.857 + 0.514i)T \)
73 \( 1 + (-0.985 + 0.168i)T \)
79 \( 1 + (-0.990 - 0.134i)T \)
83 \( 1 + (0.638 + 0.769i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.874 + 0.485i)T \)
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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

−23.94534405828800596945902711726, −23.1095534156556885522673058388, −22.2414077769817876459072745685, −21.96533247567790161044112135585, −20.953568716865464510650759403067, −19.69696270236798622176634904424, −18.9248494802946936377909830679, −18.20575451944232307660134298788, −17.30501936874241140897882028537, −16.06703969675491448918496141930, −15.30942108303064986045887481824, −13.81991395785015044678139650861, −13.311744761473937182242463261107, −12.68542944187552193170210334247, −11.43925106022283524435827134335, −10.61952708793080604394734501088, −10.18273363445209780107008434025, −8.79864888000087492616736634282, −6.98786424269926455497297466114, −6.03102105774790708183551749326, −5.834028009855464690827971044, −4.297911662496338255343466846721, −3.03837247943997341420616237618, −2.0611552593375528067796206421, −0.50186436840267689806623540154, 1.92625108906687963816016165803, 3.58605996419778453796841787641, 4.59282439389258132627536414466, 5.42054130973279964994566071605, 6.38834360777164809628057733847, 6.904002664812922560386429882247, 8.62281726431186167728304437195, 9.51834809539154958460911239424, 10.44908683063043773343695814779, 11.880327496476585603283093508321, 12.55197250548548339890524368043, 13.3062092395046507267207170657, 14.284243363723785810392015070638, 15.68847078950965766083332994356, 16.00224037414602052333674107535, 16.91066465602128180931428613096, 17.6549617962502628432109880532, 18.46419569789145843488232306573, 20.21053976248415850507426659356, 20.95088220002586165080882827443, 21.85770185430175278592862796108, 22.36915061835386781400690919258, 23.404518091998240435278911366739, 23.82593662754203917690180315038, 25.16953016693920830403387558609

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