↑

Properties

Label 1-373-373.9-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

Related objects

Downloads

Learn more

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: $\chi_{373} (9, \cdot )$
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 \( 1 + (0.151 + 0.988i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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