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Properties

Label 1-373-373.169-r0-0-0
Degree $1$
Conductor $373$
Sign $0.626 - 0.779i$
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  + (−0.994 + 0.101i)2-s + (0.528 + 0.848i)3-s + (0.979 − 0.201i)4-s + (0.151 − 0.988i)5-s + (−0.612 − 0.790i)6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (−0.440 + 0.897i)9-s + (−0.0506 + 0.998i)10-s + (−0.954 − 0.299i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (−0.250 + 0.968i)14-s + (0.918 − 0.394i)15-s + (0.918 − 0.394i)16-s + (0.979 − 0.201i)17-s + ⋯
L(s)  = 1  + (−0.994 + 0.101i)2-s + (0.528 + 0.848i)3-s + (0.979 − 0.201i)4-s + (0.151 − 0.988i)5-s + (−0.612 − 0.790i)6-s + (0.347 − 0.937i)7-s + (−0.954 + 0.299i)8-s + (−0.440 + 0.897i)9-s + (−0.0506 + 0.998i)10-s + (−0.954 − 0.299i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (−0.250 + 0.968i)14-s + (0.918 − 0.394i)15-s + (0.918 − 0.394i)16-s + (0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.626 - 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8022437909 - 0.3842197083i\)
\(L(\frac12)\) \(\approx\) \(0.8022437909 - 0.3842197083i\)
\(L(1)\) \(\approx\) \(0.8059359399 - 0.07122161243i\)
\(L(1)\) \(\approx\) \(0.8059359399 - 0.07122161243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.994 + 0.101i)T \)
3 \( 1 + (0.528 + 0.848i)T \)
5 \( 1 + (0.151 - 0.988i)T \)
7 \( 1 + (0.347 - 0.937i)T \)
11 \( 1 + (-0.954 - 0.299i)T \)
13 \( 1 + (-0.954 - 0.299i)T \)
17 \( 1 + (0.979 - 0.201i)T \)
19 \( 1 + (0.918 + 0.394i)T \)
23 \( 1 + (0.528 - 0.848i)T \)
29 \( 1 + (-0.874 + 0.485i)T \)
31 \( 1 + (0.688 - 0.724i)T \)
37 \( 1 + (0.820 - 0.571i)T \)
41 \( 1 + (-0.612 - 0.790i)T \)
43 \( 1 + (0.979 - 0.201i)T \)
47 \( 1 + (-0.250 - 0.968i)T \)
53 \( 1 + (-0.440 - 0.897i)T \)
59 \( 1 + (-0.874 + 0.485i)T \)
61 \( 1 + (0.528 + 0.848i)T \)
67 \( 1 + (0.151 - 0.988i)T \)
71 \( 1 + (0.979 + 0.201i)T \)
73 \( 1 + (-0.440 - 0.897i)T \)
79 \( 1 + (-0.0506 + 0.998i)T \)
83 \( 1 + (-0.440 - 0.897i)T \)
89 \( 1 + T \)
97 \( 1 + (0.979 - 0.201i)T \)
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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

−24.996164347452235429603849755327, −24.16615436261010758759070645738, −23.20937833757777921476593125917, −21.84494004250088355393321531391, −21.11353967552763434287919495184, −20.11460206845395937977257576697, −18.9847469333481533140984377666, −18.784411445714759886962029317173, −17.881727066792036910924077445701, −17.272944096947536191217158472097, −15.69757534922763237753579557834, −15.009951266344382038160593870217, −14.220178723486200878518914049089, −12.88048147255657131616460868924, −11.92983282623577641749965448136, −11.23509603737323163425390414914, −9.887422639850767762366727939327, −9.29186165590167714319762979086, −7.86155391820682635582682893829, −7.5917165698967914550984755554, −6.4764585341334609779701352722, −5.42819027221453232186160098555, −3.06783741451016838873587619076, −2.595921092458594176499623441602, −1.51749630769424302293912700174, 0.698319247877955598910124962037, 2.20901608878353514390015197916, 3.43485033452616822156582829748, 4.860820630002822199328285707, 5.593789885471139068720619442406, 7.51869239966747372257040140610, 7.925533512809402808820610135, 8.985655365258925540559449732383, 9.91072962187634356160511738871, 10.419506061279157558696493489295, 11.54083066745378238316243093903, 12.77159933606221486622732914937, 13.958897117470060724420752311416, 14.85915151311456006688109147881, 15.90470630541553647605756201458, 16.660806045185538940710362258569, 17.06372888362227804621209350585, 18.30365973779338767162667802803, 19.38444376783277112021385011665, 20.29002704367556970438622499507, 20.694840723724735572641240825816, 21.344303411074824099585565612880, 22.76845217708702620132181117167, 24.01838810184517968068597593931, 24.60744842231240012212818775736

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