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Properties

Label 1-373-373.184-r0-0-0
Degree $1$
Conductor $373$
Sign $0.0459 - 0.998i$
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  + (0.954 − 0.299i)2-s + (−0.994 + 0.101i)3-s + (0.820 − 0.571i)4-s + (0.440 − 0.897i)5-s + (−0.918 + 0.394i)6-s + (−0.874 + 0.485i)7-s + (0.612 − 0.790i)8-s + (0.979 − 0.201i)9-s + (0.151 − 0.988i)10-s + (0.612 + 0.790i)11-s + (−0.758 + 0.651i)12-s + (−0.612 − 0.790i)13-s + (−0.688 + 0.724i)14-s + (−0.347 + 0.937i)15-s + (0.347 − 0.937i)16-s + (0.820 − 0.571i)17-s + ⋯
L(s)  = 1  + (0.954 − 0.299i)2-s + (−0.994 + 0.101i)3-s + (0.820 − 0.571i)4-s + (0.440 − 0.897i)5-s + (−0.918 + 0.394i)6-s + (−0.874 + 0.485i)7-s + (0.612 − 0.790i)8-s + (0.979 − 0.201i)9-s + (0.151 − 0.988i)10-s + (0.612 + 0.790i)11-s + (−0.758 + 0.651i)12-s + (−0.612 − 0.790i)13-s + (−0.688 + 0.724i)14-s + (−0.347 + 0.937i)15-s + (0.347 − 0.937i)16-s + (0.820 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0459 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.0459 - 0.998i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.0459 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221820774 - 1.166939557i\)
\(L(\frac12)\) \(\approx\) \(1.221820774 - 1.166939557i\)
\(L(1)\) \(\approx\) \(1.291205311 - 0.5599431895i\)
\(L(1)\) \(\approx\) \(1.291205311 - 0.5599431895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.954 - 0.299i)T \)
3 \( 1 + (-0.994 + 0.101i)T \)
5 \( 1 + (0.440 - 0.897i)T \)
7 \( 1 + (-0.874 + 0.485i)T \)
11 \( 1 + (0.612 + 0.790i)T \)
13 \( 1 + (-0.612 - 0.790i)T \)
17 \( 1 + (0.820 - 0.571i)T \)
19 \( 1 + (-0.347 - 0.937i)T \)
23 \( 1 + (0.994 + 0.101i)T \)
29 \( 1 + (-0.0506 + 0.998i)T \)
31 \( 1 + (-0.758 - 0.651i)T \)
37 \( 1 + (-0.250 - 0.968i)T \)
41 \( 1 + (0.918 - 0.394i)T \)
43 \( 1 + (-0.820 + 0.571i)T \)
47 \( 1 + (-0.688 - 0.724i)T \)
53 \( 1 + (-0.979 - 0.201i)T \)
59 \( 1 + (-0.0506 + 0.998i)T \)
61 \( 1 + (0.994 - 0.101i)T \)
67 \( 1 + (0.440 - 0.897i)T \)
71 \( 1 + (0.820 + 0.571i)T \)
73 \( 1 + (0.979 + 0.201i)T \)
79 \( 1 + (-0.151 + 0.988i)T \)
83 \( 1 + (0.979 + 0.201i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.820 + 0.571i)T \)
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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

−24.70751044544179938527399930004, −23.60605782135065177372405388714, −23.082214712279472013416599431532, −22.25904598762914265632485238593, −21.734378035070692631761202184483, −20.925263028102293066458813898974, −19.2823443899355538805341242808, −18.88642306634091082815004064810, −17.29541265943265573270392403387, −16.82982651496528641561438693384, −16.09847208530581560666487837021, −14.85086107932347451450030873591, −14.10604496214224578869353802474, −13.16764786761457502832517923663, −12.30260346155934670007821328872, −11.36864492700599952142324406181, −10.569429604221984552639774932821, −9.64021937756857931859956872202, −7.7646098295515023692547918711, −6.64284261244887211621960050443, −6.386654361965248630095494303037, −5.36337611121637361716638842282, −4.02809275323428728537690586301, −3.18275770982035986644023362503, −1.65892789200052802741987676119, 0.88228705291280440810370339092, 2.26415574920422078200916839657, 3.66055521268954206262423735267, 4.976186136860369698727004673007, 5.33701967225629998193132323530, 6.44633044949788839910867218608, 7.29306001117767061271526745825, 9.320551565988403656829591951493, 9.85284402457823272166718583475, 11.00810473548657990943092895807, 12.1055457475409447408592931345, 12.65995161926511222372582428602, 13.156773236821143859901832883141, 14.63866033775582674457643633341, 15.52664886773353604571197245687, 16.384255378540440232831547839876, 17.08479635808824032518902396243, 18.147648000570502643925709302098, 19.41007855605410488913769314382, 20.16232552103206696148987036294, 21.190826431959581080344166223, 21.89526834938651980882992154142, 22.6571568454382540113511926444, 23.24725242392529213059505862046, 24.32176176127588519975893390329

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