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Properties

Label 1-373-373.235-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.0446 + 0.999i$
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  + (0.857 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.0446 + 0.999i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.0446 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035898679 + 1.083179164i\)
\(L(\frac12)\) \(\approx\) \(1.035898679 + 1.083179164i\)
\(L(1)\) \(\approx\) \(1.122378908 + 0.5795881353i\)
\(L(1)\) \(\approx\) \(1.122378908 + 0.5795881353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.857 + 0.514i)T \)
3 \( 1 + (-0.985 + 0.168i)T \)
5 \( 1 + (-0.972 - 0.234i)T \)
7 \( 1 + (0.979 + 0.201i)T \)
11 \( 1 + (0.890 + 0.455i)T \)
13 \( 1 + (-0.0506 - 0.998i)T \)
17 \( 1 + (0.528 - 0.848i)T \)
19 \( 1 + (-0.440 + 0.897i)T \)
23 \( 1 + (0.347 + 0.937i)T \)
29 \( 1 + (-0.905 + 0.425i)T \)
31 \( 1 + (-0.612 + 0.790i)T \)
37 \( 1 + (0.585 - 0.810i)T \)
41 \( 1 + (0.151 + 0.988i)T \)
43 \( 1 + (0.470 + 0.882i)T \)
47 \( 1 + (0.217 + 0.975i)T \)
53 \( 1 + (0.943 + 0.331i)T \)
59 \( 1 + (0.0843 - 0.996i)T \)
61 \( 1 + (-0.985 + 0.168i)T \)
67 \( 1 + (0.688 - 0.724i)T \)
71 \( 1 + (-0.999 + 0.0337i)T \)
73 \( 1 + (-0.184 - 0.982i)T \)
79 \( 1 + (-0.713 - 0.701i)T \)
83 \( 1 + (0.943 + 0.331i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.528 - 0.848i)T \)
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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

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

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