Properties

Label 1-373-373.28-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.371 - 0.928i$
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  + (−0.972 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.371 - 0.928i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.371 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5625939819 - 0.8307808888i\)
\(L(\frac12)\) \(\approx\) \(0.5625939819 - 0.8307808888i\)
\(L(1)\) \(\approx\) \(0.8066400820 - 0.4032972895i\)
\(L(1)\) \(\approx\) \(0.8066400820 - 0.4032972895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.972 - 0.234i)T \)
3 \( 1 + (0.963 - 0.266i)T \)
5 \( 1 + (0.638 - 0.769i)T \)
7 \( 1 + (-0.954 + 0.299i)T \)
11 \( 1 + (-0.184 - 0.982i)T \)
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 \( 1 + (-0.378 - 0.925i)T \)
31 \( 1 + (0.979 - 0.201i)T \)
37 \( 1 + (-0.931 - 0.363i)T \)
41 \( 1 + (0.528 + 0.848i)T \)
43 \( 1 + (0.890 + 0.455i)T \)
47 \( 1 + (-0.557 + 0.830i)T \)
53 \( 1 + (0.857 + 0.514i)T \)
59 \( 1 + (0.990 + 0.134i)T \)
61 \( 1 + (0.963 - 0.266i)T \)
67 \( 1 + (0.347 + 0.937i)T \)
71 \( 1 + (-0.839 - 0.543i)T \)
73 \( 1 + (0.0168 - 0.999i)T \)
79 \( 1 + (-0.801 + 0.598i)T \)
83 \( 1 + (0.857 + 0.514i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−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