Properties

Degree 1
Conductor 373
Sign $-0.0446 - 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.857 − 0.514i)2-s + (−0.985 − 0.168i)3-s + (0.470 − 0.882i)4-s + (−0.972 + 0.234i)5-s + (−0.931 + 0.363i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (0.943 + 0.331i)9-s + (−0.713 + 0.701i)10-s + (0.890 − 0.455i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.736 − 0.676i)14-s + (0.997 − 0.0675i)15-s + (−0.557 − 0.830i)16-s + (0.528 + 0.848i)17-s + ⋯
L(s,χ)  = 1  + (0.857 − 0.514i)2-s + (−0.985 − 0.168i)3-s + (0.470 − 0.882i)4-s + (−0.972 + 0.234i)5-s + (−0.931 + 0.363i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (0.943 + 0.331i)9-s + (−0.713 + 0.701i)10-s + (0.890 − 0.455i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.736 − 0.676i)14-s + (0.997 − 0.0675i)15-s + (−0.557 − 0.830i)16-s + (0.528 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0446 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0446 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\n\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $-0.0446 - 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (100, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ -0.0446 - 0.999i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.035898679 - 1.083179164i$
$L(\frac12,\chi)$  $\approx$  $1.035898679 - 1.083179164i$
$L(\chi,1)$  $\approx$  1.122378908 - 0.5795881353i
$L(1,\chi)$  $\approx$  1.122378908 - 0.5795881353i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.70452826398901314529045321130, −23.73622679138223205866825375840, −23.091238626509160680145394850121, −22.55357701504978754658501827214, −21.5228686369049535874275473572, −20.71972545400725440931337384307, −19.85307206119795497590720580826, −18.37104243667781627887802895621, −17.529160859052162033055330460490, −16.70162944458372878276370950832, −15.94797001102313536165321568171, −15.00613856515604423390489706232, −14.478422714221467666134285342344, −12.88889075487321639008604967723, −12.21931949612916756828334965016, −11.51902815722375227595008749430, −10.8143802589541761989712748981, −9.1767147875458032578794798548, −7.78260365784608795164937754639, −7.32899690717247164410564807785, −5.9575080601665178648856597319, −5.10989971140043356195332568573, −4.34744583941644548291679743096, −3.368514014777422821723076714630, −1.42780737631573727240148132124, 0.89664995796891060643459470420, 2.11977551212912614240448239113, 3.96015163288687644225027827547, 4.322468369113200254164117605006, 5.5253296679336689626039484782, 6.61092522976918905168732700290, 7.37044063378124131227057382828, 8.82790068507903480088724499182, 10.38836708604190023607806448954, 11.20654934323089787147482354107, 11.61795742930486023712787981499, 12.4042193800991655987809499100, 13.52780392508088671919830847607, 14.63563115093742801681141167600, 15.209496917504796912388992681889, 16.48982477736146598304907446963, 17.0938406944676851487596200231, 18.60978014614340782380076024611, 19.03461988338508910179330511394, 20.07165912087933253710920798685, 21.144676435539265817451074798401, 21.91304380558044279827076427560, 22.60124953081716949487015285964, 23.62734650348759870109297895117, 23.988581000646525830649425309692

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