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Properties

Label 1-373-373.167-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.520 + 0.853i$
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  + (−0.724 + 0.688i)2-s + (0.250 + 0.968i)3-s + (0.0506 − 0.998i)4-s + (0.937 + 0.347i)5-s + (−0.848 − 0.528i)6-s + (0.954 + 0.299i)7-s + (0.651 + 0.758i)8-s + (−0.874 + 0.485i)9-s + (−0.918 + 0.394i)10-s + (0.651 − 0.758i)11-s + (0.979 − 0.201i)12-s + (0.758 + 0.651i)13-s + (−0.897 + 0.440i)14-s + (−0.101 + 0.994i)15-s + (−0.994 − 0.101i)16-s + (0.0506 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.724 + 0.688i)2-s + (0.250 + 0.968i)3-s + (0.0506 − 0.998i)4-s + (0.937 + 0.347i)5-s + (−0.848 − 0.528i)6-s + (0.954 + 0.299i)7-s + (0.651 + 0.758i)8-s + (−0.874 + 0.485i)9-s + (−0.918 + 0.394i)10-s + (0.651 − 0.758i)11-s + (0.979 − 0.201i)12-s + (0.758 + 0.651i)13-s + (−0.897 + 0.440i)14-s + (−0.101 + 0.994i)15-s + (−0.994 − 0.101i)16-s + (0.0506 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.520 + 0.853i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.520 + 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097750477 + 1.955778312i\)
\(L(\frac12)\) \(\approx\) \(1.097750477 + 1.955778312i\)
\(L(1)\) \(\approx\) \(0.9327111849 + 0.7524786203i\)
\(L(1)\) \(\approx\) \(0.9327111849 + 0.7524786203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.724 + 0.688i)T \)
3 \( 1 + (0.250 + 0.968i)T \)
5 \( 1 + (0.937 + 0.347i)T \)
7 \( 1 + (0.954 + 0.299i)T \)
11 \( 1 + (0.651 - 0.758i)T \)
13 \( 1 + (0.758 + 0.651i)T \)
17 \( 1 + (0.0506 - 0.998i)T \)
19 \( 1 + (0.101 + 0.994i)T \)
23 \( 1 + (0.968 + 0.250i)T \)
29 \( 1 + (-0.612 - 0.790i)T \)
31 \( 1 + (-0.979 - 0.201i)T \)
37 \( 1 + (-0.151 + 0.988i)T \)
41 \( 1 + (0.528 - 0.848i)T \)
43 \( 1 + (0.998 + 0.0506i)T \)
47 \( 1 + (0.897 + 0.440i)T \)
53 \( 1 + (0.485 - 0.874i)T \)
59 \( 1 + (0.612 + 0.790i)T \)
61 \( 1 + (0.968 - 0.250i)T \)
67 \( 1 + (0.937 + 0.347i)T \)
71 \( 1 + (0.0506 + 0.998i)T \)
73 \( 1 + (-0.874 - 0.485i)T \)
79 \( 1 + (0.394 + 0.918i)T \)
83 \( 1 + (-0.874 - 0.485i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.998 - 0.0506i)T \)
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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

−24.3576557335698696619186042711, −23.309692455450028730723956016537, −22.1387610008563957499608844723, −21.17943228421766135984579016935, −20.35983578031803246269072688872, −19.88063237151864611600126848088, −18.70720831710249856143877580672, −17.72179217153439939411768961834, −17.59523239692641190241832790736, −16.66275336990648341314067737279, −14.97657126967402243908817877880, −14.01916886848340559797654907192, −12.970716969592225394245782067346, −12.58879892860496999741948803085, −11.270372458510031503296875229644, −10.60170554615629506228083652653, −9.1823499529656544439693845295, −8.69728308464888037178111079889, −7.59259606224697175017135614351, −6.7405881500840381705204056300, −5.39393848129400549027612511711, −3.935733098489541716555383506212, −2.50556386219752894052603520350, −1.57689994205201746659314921918, −0.92234323009981533349208167572, 1.23706284124965249367545776519, 2.43515552533639970540962198741, 4.0095557262592118019024687263, 5.362279479109410661221136105304, 5.861267977948978861908743375887, 7.17446197115919642756583258809, 8.465815765775793465370175440517, 9.09549432506486944876489209089, 9.84106800515203972994523809726, 10.996870858202913355542336231119, 11.42523186571813288187261529927, 13.66845724858731557214647904908, 14.23320372327372673456110163168, 14.85778339807060484054927151042, 15.94023390377801583019549725434, 16.762849064150531379497990266303, 17.41764436073574017131532735866, 18.49790433291081769300888491902, 19.092796212005281710448998053537, 20.65095943124829395066030509733, 20.93036680669455593684987319846, 22.073210470727264801780575633740, 22.868246519786965975241241420701, 24.13838954936929735139705080296, 24.9856727110809324204780046697

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