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Properties

Label 1-373-373.83-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.328 + 0.944i$
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  + (−0.184 − 0.982i)2-s + (−0.972 + 0.234i)3-s + (−0.931 + 0.363i)4-s + (0.0168 − 0.999i)5-s + (0.409 + 0.912i)6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.890 − 0.455i)9-s + (−0.985 + 0.168i)10-s + (−0.999 − 0.0337i)11-s + (0.820 − 0.571i)12-s + (0.528 − 0.848i)13-s + (−0.664 + 0.747i)14-s + (0.217 + 0.975i)15-s + (0.736 − 0.676i)16-s + (0.151 − 0.988i)17-s + ⋯
L(s)  = 1  + (−0.184 − 0.982i)2-s + (−0.972 + 0.234i)3-s + (−0.931 + 0.363i)4-s + (0.0168 − 0.999i)5-s + (0.409 + 0.912i)6-s + (−0.612 − 0.790i)7-s + (0.528 + 0.848i)8-s + (0.890 − 0.455i)9-s + (−0.985 + 0.168i)10-s + (−0.999 − 0.0337i)11-s + (0.820 − 0.571i)12-s + (0.528 − 0.848i)13-s + (−0.664 + 0.747i)14-s + (0.217 + 0.975i)15-s + (0.736 − 0.676i)16-s + (0.151 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1588975660 - 0.2235488972i\)
\(L(\frac12)\) \(\approx\) \(-0.1588975660 - 0.2235488972i\)
\(L(1)\) \(\approx\) \(0.3491290786 - 0.3642594083i\)
\(L(1)\) \(\approx\) \(0.3491290786 - 0.3642594083i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.184 - 0.982i)T \)
3 \( 1 + (-0.972 + 0.234i)T \)
5 \( 1 + (0.0168 - 0.999i)T \)
7 \( 1 + (-0.612 - 0.790i)T \)
11 \( 1 + (-0.999 - 0.0337i)T \)
13 \( 1 + (0.528 - 0.848i)T \)
17 \( 1 + (0.151 - 0.988i)T \)
19 \( 1 + (-0.954 + 0.299i)T \)
23 \( 1 + (0.688 - 0.724i)T \)
29 \( 1 + (-0.801 - 0.598i)T \)
31 \( 1 + (0.820 + 0.571i)T \)
37 \( 1 + (0.997 - 0.0675i)T \)
41 \( 1 + (-0.994 - 0.101i)T \)
43 \( 1 + (-0.931 + 0.363i)T \)
47 \( 1 + (-0.315 + 0.948i)T \)
53 \( 1 + (0.890 + 0.455i)T \)
59 \( 1 + (-0.117 + 0.993i)T \)
61 \( 1 + (-0.972 + 0.234i)T \)
67 \( 1 + (-0.874 + 0.485i)T \)
71 \( 1 + (0.780 - 0.625i)T \)
73 \( 1 + (-0.839 + 0.543i)T \)
79 \( 1 + (-0.985 + 0.168i)T \)
83 \( 1 + (0.890 + 0.455i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.151 - 0.988i)T \)
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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

−25.40232960235780055696583675765, −24.169258153768333940512588011631, −23.377153264788750442144994885117, −22.986390389866937869500312851304, −21.73232659201907541919790065021, −21.59803611736371557192444147221, −19.3161366924472598247449853756, −18.7003891644988657730784948110, −18.26499789327460633945655667662, −17.1998303832841558688643221836, −16.42518762114693785891295613290, −15.38921838080353415917314038534, −15.003339095618726156773051605145, −13.45984096059498504670788073764, −12.93637691898919656525425095308, −11.60149224273783380920711343503, −10.60383165815069655849336403648, −9.82135419180529650275109514873, −8.569512400509746993624916989016, −7.42751959161970013543157432759, −6.49837805329650482410051820001, −6.01102971976970153483906015863, −4.96943999625610469997427815062, −3.60714961581732828030551154097, −1.93551526721555846757750892338, 0.22657503930228280422355675631, 1.19796927208997938490208499685, 2.938755964675067245589912652498, 4.20624084797743279329736226682, 4.93405506038384017262568274218, 5.96350549980500413946585973691, 7.4996755407784757970949640350, 8.57700990308294892343724667069, 9.76347389430136334221111244041, 10.37895805423813826136750335653, 11.20624931127275795126177700352, 12.30658767630094146460202354784, 13.01755145964653131193608750303, 13.51362605220012883393432617706, 15.32358143144926840425577437090, 16.44455561083691657722160849854, 16.87399825809908997402160891798, 17.89089143133167637929583958977, 18.65854384959998505985825755508, 19.77907205865878514604859687384, 20.84584520647800726349888649728, 20.946840436702297664989423850899, 22.29050139269513441570619502984, 23.23103066404479735951574493867, 23.38377262741843899126794403392

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