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Properties

Label 1-373-373.44-r1-0-0
Degree $1$
Conductor $373$
Sign $0.307 - 0.951i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (−0.514 − 0.857i)2-s + (0.985 + 0.168i)3-s + (−0.470 + 0.882i)4-s + (0.234 + 0.972i)5-s + (−0.363 − 0.931i)6-s + (−0.979 + 0.201i)7-s + (0.998 − 0.0506i)8-s + (0.943 + 0.331i)9-s + (0.713 − 0.701i)10-s + (−0.455 − 0.890i)11-s + (−0.612 + 0.790i)12-s + (0.0506 − 0.998i)13-s + (0.676 + 0.736i)14-s + (0.0675 + 0.997i)15-s + (−0.557 − 0.830i)16-s + (−0.528 − 0.848i)17-s + ⋯
L(s)  = 1  + (−0.514 − 0.857i)2-s + (0.985 + 0.168i)3-s + (−0.470 + 0.882i)4-s + (0.234 + 0.972i)5-s + (−0.363 − 0.931i)6-s + (−0.979 + 0.201i)7-s + (0.998 − 0.0506i)8-s + (0.943 + 0.331i)9-s + (0.713 − 0.701i)10-s + (−0.455 − 0.890i)11-s + (−0.612 + 0.790i)12-s + (0.0506 − 0.998i)13-s + (0.676 + 0.736i)14-s + (0.0675 + 0.997i)15-s + (−0.557 − 0.830i)16-s + (−0.528 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425382548 - 1.037785450i\)
\(L(\frac12)\) \(\approx\) \(1.425382548 - 1.037785450i\)
\(L(1)\) \(\approx\) \(1.032806365 - 0.3035924030i\)
\(L(1)\) \(\approx\) \(1.032806365 - 0.3035924030i\)

Euler product

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

−24.69310965968803620932257016385, −24.02107122840227252874064701155, −23.17885642449596219306251748370, −22.01090430819960330656586056035, −20.69087845275078944782342702787, −20.1261502936753116093213414410, −19.18273481406170062880673061670, −18.5407992164247942753573967840, −17.32851796154860985063994549474, −16.550793006478217825084457669577, −15.703326288209372387722796139864, −14.96967678416941969474770899923, −13.78821068515873733661306129205, −13.2044274154785224544342972954, −12.33909791979882873704649194387, −10.44059482264235452594304989048, −9.44540409016526061521897439479, −9.13208949471226221140354091473, −8.00998645604772904652608740413, −7.13301399014212481335883340073, −6.19883312581390062140888722428, −4.83315372738303452794536396894, −3.90437204180651860242575105784, −2.169338635944038967139427272629, −1.06540016642300507223263365852, 0.600742428844297424288191086046, 2.38997232147182983509871611907, 3.03026703561699332171421279687, 3.61694318925918332138153242412, 5.349732180231018917076716718860, 6.958236588514019462034297476497, 7.72865008918292515386281788114, 8.94043456729801715247854316074, 9.583422212860504581118918062272, 10.48947192229432764904242804794, 11.21907295551311580449833799897, 12.65308003596456672805705521120, 13.46315942060208626082208131235, 14.03586781821203537431098358657, 15.50700904281136364091701884298, 16.00381663555399576960616374375, 17.49993511654813327656216079301, 18.41808675646871405137270340935, 19.04404182926046492542883515561, 19.68970966079987542663035682416, 20.643556003180069325500190246235, 21.4477004195608371954857410014, 22.33405542236011263113161830620, 22.82656073739347507946511038690, 24.59755488028853705546320472878

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