Properties

Degree 1
Conductor 373
Sign $-0.371 - 0.928i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.972 − 0.234i)2-s + (0.963 − 0.266i)3-s + (0.890 + 0.455i)4-s + (0.638 − 0.769i)5-s + (−0.999 + 0.0337i)6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (0.857 − 0.514i)9-s + (−0.801 + 0.598i)10-s + (−0.184 − 0.982i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (0.997 − 0.0675i)14-s + (0.409 − 0.912i)15-s + (0.585 + 0.810i)16-s + (−0.0506 − 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.972 − 0.234i)2-s + (0.963 − 0.266i)3-s + (0.890 + 0.455i)4-s + (0.638 − 0.769i)5-s + (−0.999 + 0.0337i)6-s + (−0.954 + 0.299i)7-s + (−0.758 − 0.651i)8-s + (0.857 − 0.514i)9-s + (−0.801 + 0.598i)10-s + (−0.184 − 0.982i)11-s + (0.979 + 0.201i)12-s + (−0.758 + 0.651i)13-s + (0.997 − 0.0675i)14-s + (0.409 − 0.912i)15-s + (0.585 + 0.810i)16-s + (−0.0506 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.371 - 0.928i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.371 - 0.928i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $-0.371 - 0.928i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (28, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 373,\ (0:\ ),\ -0.371 - 0.928i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5625939819 - 0.8307808888i$
$L(\frac12,\chi)$  $\approx$  $0.5625939819 - 0.8307808888i$
$L(\chi,1)$  $\approx$  0.8066400820 - 0.4032972895i
$L(1,\chi)$  $\approx$  0.8066400820 - 0.4032972895i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−25.41971687307413479937539454913, −24.394912667535620850202312960507, −23.189312103882920291213267599861, −22.12546284625136907965306155134, −21.19538418188912108258948811701, −20.25427175800977940733384563457, −19.41622868454853514338167175661, −18.99443235426082457504099566291, −17.73048607813985324542441853237, −17.16965518495626258724451932544, −15.88926694581470451412273241557, −15.11149417364784200659922043582, −14.5506466135066437897268613060, −13.319852489292907121781972720811, −12.39267464141216949774980569137, −10.58834519917088678502057083549, −10.17523024556910126338965742636, −9.52550496436645419946531453562, −8.45286123454796881718131297196, −7.313325256706841620008044881747, −6.77314626761328380344010904248, −5.47847619939152024878435009402, −3.72226026427047602501896391454, −2.62640367250226879699233031218, −1.82659223838429174759885673379, 0.69367865085001573546010738330, 2.233875813514548360523058107514, 2.790694591172800662748392739284, 4.25292047058516562441484215170, 6.029828384400783106023494604571, 6.85697321413669779516889580941, 8.09632392334013439489015811348, 8.89465409905798635784385951735, 9.49050558353988302603630924964, 10.26050018096704352266044581820, 11.77663764327680159611183165791, 12.67917861344117624336785331734, 13.39951462362496600751355281751, 14.484750981147485078640364798598, 15.804159139572590640147981891646, 16.36534478834635259636969398428, 17.31182041635329251111905566090, 18.43044326114406048531887255863, 19.19875435869307564235968940403, 19.66272108380268067870881983091, 20.8673784766013799327881365064, 21.201665207239377912155655489556, 22.29515150290884104382584306312, 24.0208334521946003554723890377, 24.70239879908432268807812693748

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