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Properties

Label 1-373-373.63-r0-0-0
Degree $1$
Conductor $373$
Sign $0.996 - 0.0880i$
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  + (0.378 − 0.925i)2-s + (0.217 + 0.975i)3-s + (−0.713 − 0.701i)4-s + (0.315 + 0.948i)5-s + (0.985 + 0.168i)6-s + (−0.0506 − 0.998i)7-s + (−0.918 + 0.394i)8-s + (−0.905 + 0.425i)9-s + (0.997 + 0.0675i)10-s + (0.801 − 0.598i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.943 − 0.331i)14-s + (−0.857 + 0.514i)15-s + (0.0168 + 0.999i)16-s + (−0.250 + 0.968i)17-s + ⋯
L(s)  = 1  + (0.378 − 0.925i)2-s + (0.217 + 0.975i)3-s + (−0.713 − 0.701i)4-s + (0.315 + 0.948i)5-s + (0.985 + 0.168i)6-s + (−0.0506 − 0.998i)7-s + (−0.918 + 0.394i)8-s + (−0.905 + 0.425i)9-s + (0.997 + 0.0675i)10-s + (0.801 − 0.598i)11-s + (0.528 − 0.848i)12-s + (0.918 + 0.394i)13-s + (−0.943 − 0.331i)14-s + (−0.857 + 0.514i)15-s + (0.0168 + 0.999i)16-s + (−0.250 + 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.996 - 0.0880i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.996 - 0.0880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.680495181 - 0.07416206856i\)
\(L(\frac12)\) \(\approx\) \(1.680495181 - 0.07416206856i\)
\(L(1)\) \(\approx\) \(1.359408546 - 0.1516711809i\)
\(L(1)\) \(\approx\) \(1.359408546 - 0.1516711809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.378 - 0.925i)T \)
3 \( 1 + (0.217 + 0.975i)T \)
5 \( 1 + (0.315 + 0.948i)T \)
7 \( 1 + (-0.0506 - 0.998i)T \)
11 \( 1 + (0.801 - 0.598i)T \)
13 \( 1 + (0.918 + 0.394i)T \)
17 \( 1 + (-0.250 + 0.968i)T \)
19 \( 1 + (0.874 - 0.485i)T \)
23 \( 1 + (0.954 - 0.299i)T \)
29 \( 1 + (-0.931 - 0.363i)T \)
31 \( 1 + (0.528 + 0.848i)T \)
37 \( 1 + (0.283 + 0.959i)T \)
41 \( 1 + (0.347 + 0.937i)T \)
43 \( 1 + (0.713 + 0.701i)T \)
47 \( 1 + (0.184 - 0.982i)T \)
53 \( 1 + (0.905 + 0.425i)T \)
59 \( 1 + (0.780 - 0.625i)T \)
61 \( 1 + (-0.217 - 0.975i)T \)
67 \( 1 + (-0.979 - 0.201i)T \)
71 \( 1 + (0.963 + 0.266i)T \)
73 \( 1 + (0.0843 + 0.996i)T \)
79 \( 1 + (-0.997 - 0.0675i)T \)
83 \( 1 + (-0.905 - 0.425i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.250 - 0.968i)T \)
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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

−24.54002536885204213406072188815, −24.204442745059720530406598718558, −22.841451538310497781512193332393, −22.50155168953412786657134186349, −21.03799256323741856932361935664, −20.43637050512487930318460630920, −19.14304111728301976121684813510, −18.16133609441922896288456206886, −17.64031820948728924905483412834, −16.63239227515042838120697064876, −15.72500143714127013831551752397, −14.77362014699539361503757281434, −13.79294863279254143986822057205, −13.0772091946555188428867266583, −12.30066485191884270004291524448, −11.61561653499846276503116888163, −9.20091514241599393508105514257, −9.10152749244116866714485985869, −7.926217010249902325499840088500, −7.01978439683302902076877467613, −5.848999143381897017616126073339, −5.37691408278820198495214702238, −3.91323364962090164614471069667, −2.54290153297011775688231129777, −1.09296013663588133953687155800, 1.34172167016428290520968905539, 2.9129883829827887887642279072, 3.63128105568197452289456505189, 4.39695065634769121478912627129, 5.76361553010499897408493351293, 6.721012352955740537940599851452, 8.42212355531951386354364571904, 9.42035678539878200171997093702, 10.2151363129157477381147630678, 11.109292874211803180468267740791, 11.37927581893062854692247729096, 13.21214023389553664777525139061, 13.89266728017544696438171984666, 14.53408902854229153117154607313, 15.43128916700089519867699832466, 16.708455126644642651628801678857, 17.55985510824682390992923249436, 18.739201423089782163291065040496, 19.563844834079787939325685791171, 20.294144880648175148966031886281, 21.27917192085346623472196092776, 21.79465436933913907335269372837, 22.708242429775747152314836117022, 23.217720197409716653159099419492, 24.45745943061300145990440238737

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