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Properties

Label 1-373-373.55-r0-0-0
Degree $1$
Conductor $373$
Sign $0.406 + 0.913i$
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  + (0.758 + 0.651i)2-s + (0.688 − 0.724i)3-s + (0.151 + 0.988i)4-s + (0.874 + 0.485i)5-s + (0.994 − 0.101i)6-s + (−0.612 + 0.790i)7-s + (−0.528 + 0.848i)8-s + (−0.0506 − 0.998i)9-s + (0.347 + 0.937i)10-s + (−0.528 − 0.848i)11-s + (0.820 + 0.571i)12-s + (0.528 + 0.848i)13-s + (−0.979 + 0.201i)14-s + (0.954 − 0.299i)15-s + (−0.954 + 0.299i)16-s + (0.151 + 0.988i)17-s + ⋯
L(s)  = 1  + (0.758 + 0.651i)2-s + (0.688 − 0.724i)3-s + (0.151 + 0.988i)4-s + (0.874 + 0.485i)5-s + (0.994 − 0.101i)6-s + (−0.612 + 0.790i)7-s + (−0.528 + 0.848i)8-s + (−0.0506 − 0.998i)9-s + (0.347 + 0.937i)10-s + (−0.528 − 0.848i)11-s + (0.820 + 0.571i)12-s + (0.528 + 0.848i)13-s + (−0.979 + 0.201i)14-s + (0.954 − 0.299i)15-s + (−0.954 + 0.299i)16-s + (0.151 + 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.406 + 0.913i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.406 + 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.164470090 + 1.405184063i\)
\(L(\frac12)\) \(\approx\) \(2.164470090 + 1.405184063i\)
\(L(1)\) \(\approx\) \(1.852460341 + 0.7237261085i\)
\(L(1)\) \(\approx\) \(1.852460341 + 0.7237261085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.758 + 0.651i)T \)
3 \( 1 + (0.688 - 0.724i)T \)
5 \( 1 + (0.874 + 0.485i)T \)
7 \( 1 + (-0.612 + 0.790i)T \)
11 \( 1 + (-0.528 - 0.848i)T \)
13 \( 1 + (0.528 + 0.848i)T \)
17 \( 1 + (0.151 + 0.988i)T \)
19 \( 1 + (0.954 + 0.299i)T \)
23 \( 1 + (-0.688 - 0.724i)T \)
29 \( 1 + (0.918 + 0.394i)T \)
31 \( 1 + (0.820 - 0.571i)T \)
37 \( 1 + (-0.440 - 0.897i)T \)
41 \( 1 + (-0.994 + 0.101i)T \)
43 \( 1 + (-0.151 - 0.988i)T \)
47 \( 1 + (-0.979 - 0.201i)T \)
53 \( 1 + (0.0506 - 0.998i)T \)
59 \( 1 + (0.918 + 0.394i)T \)
61 \( 1 + (-0.688 + 0.724i)T \)
67 \( 1 + (0.874 + 0.485i)T \)
71 \( 1 + (0.151 - 0.988i)T \)
73 \( 1 + (-0.0506 + 0.998i)T \)
79 \( 1 + (-0.347 - 0.937i)T \)
83 \( 1 + (-0.0506 + 0.998i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.151 - 0.988i)T \)
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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

−24.54421565840145790995461830143, −23.23017420607741954616711864527, −22.61764232154681075992292970752, −21.69801668041251794566784264780, −20.81882255694301044765877428271, −20.27698476157524954976876368427, −19.7930682021843829498199091760, −18.42327452913482210769520208572, −17.42180919001667023566482319996, −15.97729800620658911421939786242, −15.66076904180798176358354106692, −14.29317452560797872153649076236, −13.57182421556524142998957397191, −13.16099660345924266612893743910, −11.923572277621272032236176322254, −10.52739201874410560727253313888, −9.941856921575535842407252153207, −9.43194427241687374609102988424, −7.95053220137358092875242295995, −6.59016206925209353694737015948, −5.26104262324685121399713107016, −4.68331470219695145034674500793, −3.390004517517402217013064705214, −2.64798077560557056387464039840, −1.26132158387371804678224639465, 1.92899959046302807757623373517, 2.873840264866696367337307195680, 3.69497144288981391102576365508, 5.50274635124240144576434255720, 6.23370965727236599405583180145, 6.85579311705045804033466209207, 8.23169527148251264872912856177, 8.82812392177849050978771033374, 10.07212399517556145660385944002, 11.60103498666622989514954669765, 12.50369324613328635413498621431, 13.407369174914700762204833650805, 13.94091584015257471923463850561, 14.73899914610241425834336088364, 15.74639180391677134722919486502, 16.61523718485193022266929986964, 17.88016188659762208956043377323, 18.500175749846143239035118907461, 19.33438999391360483554734898561, 20.737476847298039984102663375950, 21.411817571642094730675076446264, 22.13090134883614712175882975649, 23.13831345630743003241517428310, 24.139726702289718258500018648345, 24.690162265078176074008461386408

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