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Properties

Label 1-373-373.119-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.799 + 0.600i$
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  + (0.688 − 0.724i)2-s + (−0.250 + 0.968i)3-s + (−0.0506 − 0.998i)4-s + (0.347 + 0.937i)5-s + (0.528 + 0.848i)6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.874 − 0.485i)9-s + (0.918 + 0.394i)10-s + (−0.758 + 0.651i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (−0.440 + 0.897i)14-s + (−0.994 + 0.101i)15-s + (−0.994 + 0.101i)16-s + (−0.0506 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.688 − 0.724i)2-s + (−0.250 + 0.968i)3-s + (−0.0506 − 0.998i)4-s + (0.347 + 0.937i)5-s + (0.528 + 0.848i)6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (−0.874 − 0.485i)9-s + (0.918 + 0.394i)10-s + (−0.758 + 0.651i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (−0.440 + 0.897i)14-s + (−0.994 + 0.101i)15-s + (−0.994 + 0.101i)16-s + (−0.0506 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.799 + 0.600i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.799 + 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1619116824 + 0.4854998607i\)
\(L(\frac12)\) \(\approx\) \(0.1619116824 + 0.4854998607i\)
\(L(1)\) \(\approx\) \(0.8799396898 + 0.1219919574i\)
\(L(1)\) \(\approx\) \(0.8799396898 + 0.1219919574i\)

Euler product

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

−24.22661589325906220133585229028, −23.54149469491021258651434301164, −22.82715670913678541416989761750, −21.80987957960873797001489032292, −20.97607866758470218249289186651, −19.78184707568774015694326889117, −19.08457247373768037407735936207, −17.60690498827702658341043409336, −17.21468045687168748307628662108, −16.33801107067835718516990430034, −15.48275410265618608422918930652, −14.18036317756014325122953899891, −13.22251521898992383682261560066, −12.882476801793209416084592830491, −12.19518796984863819095365592772, −10.81825682859266355407439272163, −9.40049309797714087205247544521, −8.17984773210066369621016446646, −7.65828741386348785679557810505, −6.24318651307154621190512430126, −5.858212929094304636693106459250, −4.734918963276942786267276338422, −3.34578610560207491304485025630, −2.13100498223509366968009658077, −0.22546103464451213803064984583, 2.4159609493458687157025775267, 2.89887234789081398231247043874, 4.17949581689967973281996857785, 5.07669387398682913630870819148, 6.17109199827383086016167135880, 6.955596241948452699215217171, 9.005662740664678883043378850119, 9.96276019674739644940347236398, 10.27542276456405898401398029067, 11.37215543214264312616327293258, 12.24333598922758108063543533684, 13.2888512940660187906964636321, 14.3562378389256851349297500421, 15.05055164684681408845142658677, 15.77994330410951435193974252723, 16.88239802721787655410550041496, 18.18621682554907402143176453594, 18.89049803465189435829135530768, 19.91364450131264356977564913654, 20.803406568640912377944127325925, 21.64612435988972693318061225573, 22.27665999318646871991168825351, 22.83574128398316091161542532868, 23.60573909770453727125547302169, 25.029377771159504938139274915485

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