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Properties

Label 1-373-373.112-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.0842 - 0.996i$
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  + (−0.963 − 0.266i)2-s + (0.0843 + 0.996i)3-s + (0.857 + 0.514i)4-s + (0.117 − 0.993i)5-s + (0.184 − 0.982i)6-s + (−0.994 − 0.101i)7-s + (−0.688 − 0.724i)8-s + (−0.985 + 0.168i)9-s + (−0.378 + 0.925i)10-s + (0.972 + 0.234i)11-s + (−0.440 + 0.897i)12-s + (0.688 − 0.724i)13-s + (0.931 + 0.363i)14-s + (0.999 + 0.0337i)15-s + (0.470 + 0.882i)16-s + (−0.874 + 0.485i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (0.0843 + 0.996i)3-s + (0.857 + 0.514i)4-s + (0.117 − 0.993i)5-s + (0.184 − 0.982i)6-s + (−0.994 − 0.101i)7-s + (−0.688 − 0.724i)8-s + (−0.985 + 0.168i)9-s + (−0.378 + 0.925i)10-s + (0.972 + 0.234i)11-s + (−0.440 + 0.897i)12-s + (0.688 − 0.724i)13-s + (0.931 + 0.363i)14-s + (0.999 + 0.0337i)15-s + (0.470 + 0.882i)16-s + (−0.874 + 0.485i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0842 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0842 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.0842 - 0.996i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.0842 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3338016108 - 0.3632060727i\)
\(L(\frac12)\) \(\approx\) \(0.3338016108 - 0.3632060727i\)
\(L(1)\) \(\approx\) \(0.5845086836 - 0.07876857567i\)
\(L(1)\) \(\approx\) \(0.5845086836 - 0.07876857567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.963 - 0.266i)T \)
3 \( 1 + (0.0843 + 0.996i)T \)
5 \( 1 + (0.117 - 0.993i)T \)
7 \( 1 + (-0.994 - 0.101i)T \)
11 \( 1 + (0.972 + 0.234i)T \)
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 \( 1 + (0.217 + 0.975i)T \)
31 \( 1 + (-0.440 - 0.897i)T \)
37 \( 1 + (0.890 - 0.455i)T \)
41 \( 1 + (-0.758 - 0.651i)T \)
43 \( 1 + (-0.857 - 0.514i)T \)
47 \( 1 + (-0.780 - 0.625i)T \)
53 \( 1 + (0.985 + 0.168i)T \)
59 \( 1 + (0.736 - 0.676i)T \)
61 \( 1 + (-0.0843 - 0.996i)T \)
67 \( 1 + (-0.918 + 0.394i)T \)
71 \( 1 + (0.0168 + 0.999i)T \)
73 \( 1 + (0.638 - 0.769i)T \)
79 \( 1 + (0.378 - 0.925i)T \)
83 \( 1 + (-0.985 - 0.168i)T \)
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

−25.28488846882781049461496359385, −24.14051081385026169825587595752, −23.22063052363867420637149468579, −22.47527502351090614677420493551, −21.28020262791855643172162161666, −19.704139918386736640907607150933, −19.586695344867801174416245752086, −18.49527869761734051267325159043, −18.145914099594514886342612828860, −17.0045453426694951504581436404, −16.21109693999734719479843163332, −15.07088341159784462878849951979, −14.161308682915836265348416887485, −13.3442528202259613029328955530, −11.86415196840349064727453053364, −11.381406112727672774710228407859, −10.13188876836709942683916981506, −9.21813163435770349035744031787, −8.29164133643914996693002822202, −7.13615851726978509938987576485, −6.39109782186824736041610424160, −6.084931250678251603796337623002, −3.616240116827606257704991116074, −2.5023290710767623841117172324, −1.46826225113522519472191584590, 0.396842972523333942502398564032, 2.0735855162103100403514852924, 3.47040934015441306522140213163, 4.27579637599681483262461604369, 5.808803997441780572281205328237, 6.73895855650444751257183027764, 8.38316055048272719523950605381, 8.891632374404466970212884637687, 9.683734957013780312807066149392, 10.494793780943466637489560896, 11.48210905597171711803124793005, 12.558451128586266363655806054726, 13.39956368694931719455748366118, 15.01298956152285319150529058018, 15.781929096616649303578337310015, 16.53075254019140900991176259661, 17.1093054396301557186751797283, 18.04594978432418617064507741567, 19.51111350583057527440185418400, 20.038107518350392242524168838401, 20.5011412377205541340307243627, 21.75891791639892302054009997283, 22.20131254155296301735545376834, 23.55698131136145064614004800267, 24.7625355466745017422693585630

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