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Properties

Label 1-869-869.234-r0-0-0
Degree $1$
Conductor $869$
Sign $-0.723 - 0.690i$
Analytic cond. $4.03561$
Root an. cond. $4.03561$
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  + (0.594 − 0.804i)2-s + (0.853 − 0.520i)3-s + (−0.293 − 0.955i)4-s + (0.999 − 0.0322i)5-s + (0.0884 − 0.996i)6-s + (0.184 − 0.982i)7-s + (−0.943 − 0.331i)8-s + (0.457 − 0.889i)9-s + (0.568 − 0.822i)10-s + (−0.748 − 0.663i)12-s + (0.0884 + 0.996i)13-s + (−0.681 − 0.732i)14-s + (0.836 − 0.548i)15-s + (−0.827 + 0.561i)16-s + (0.981 − 0.192i)17-s + (−0.443 − 0.896i)18-s + ⋯
L(s)  = 1  + (0.594 − 0.804i)2-s + (0.853 − 0.520i)3-s + (−0.293 − 0.955i)4-s + (0.999 − 0.0322i)5-s + (0.0884 − 0.996i)6-s + (0.184 − 0.982i)7-s + (−0.943 − 0.331i)8-s + (0.457 − 0.889i)9-s + (0.568 − 0.822i)10-s + (−0.748 − 0.663i)12-s + (0.0884 + 0.996i)13-s + (−0.681 − 0.732i)14-s + (0.836 − 0.548i)15-s + (−0.827 + 0.561i)16-s + (0.981 − 0.192i)17-s + (−0.443 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(869\)    =    \(11 \cdot 79\)
Sign: $-0.723 - 0.690i$
Analytic conductor: \(4.03561\)
Root analytic conductor: \(4.03561\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{869} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 869,\ (0:\ ),\ -0.723 - 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201725744 - 2.998202027i\)
\(L(\frac12)\) \(\approx\) \(1.201725744 - 2.998202027i\)
\(L(1)\) \(\approx\) \(1.506351958 - 1.516257845i\)
\(L(1)\) \(\approx\) \(1.506351958 - 1.516257845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.594 - 0.804i)T \)
3 \( 1 + (0.853 - 0.520i)T \)
5 \( 1 + (0.999 - 0.0322i)T \)
7 \( 1 + (0.184 - 0.982i)T \)
13 \( 1 + (0.0884 + 0.996i)T \)
17 \( 1 + (0.981 - 0.192i)T \)
19 \( 1 + (0.369 - 0.929i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.0563 + 0.998i)T \)
31 \( 1 + (-0.993 - 0.112i)T \)
37 \( 1 + (-0.554 + 0.832i)T \)
41 \( 1 + (0.644 - 0.764i)T \)
43 \( 1 + (0.692 + 0.721i)T \)
47 \( 1 + (-0.554 - 0.832i)T \)
53 \( 1 + (-0.231 + 0.972i)T \)
59 \( 1 + (-0.999 + 0.0161i)T \)
61 \( 1 + (-0.168 + 0.985i)T \)
67 \( 1 + (-0.748 - 0.663i)T \)
71 \( 1 + (-0.607 + 0.794i)T \)
73 \( 1 + (0.513 + 0.857i)T \)
83 \( 1 + (-0.324 + 0.945i)T \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (0.715 - 0.698i)T \)
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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

−22.329489990980989329589964670319, −21.52357807319551227398656898533, −20.97025417684930003425564375054, −20.40558925878602718971256191293, −18.99438687321407233189786629901, −18.246947332514231674825903334694, −17.50385061001745595659959479852, −16.4292811212447229326713521068, −15.89867389668726963473221458061, −14.803208493159482229190194118149, −14.55624864499143094141337479797, −13.728692407117532434201599141924, −12.79697593689139692299545682449, −12.24422874883791053187438770839, −10.755282070818776912221996418932, −9.78732052155323082498266404150, −9.09631594800187987228067931952, −8.18651758666780251307219641048, −7.62949944047199567943453455502, −6.13886502317241992640548582865, −5.6141604226170367809969881789, −4.80743774036254986675527131396, −3.58025608493088562148987937913, −2.80717386984336415099431301151, −1.88551626395456955444707845796, 1.17956601112420698460043231254, 1.708189898746078429623136304144, 2.81823359711857951541936328887, 3.645337304867855368568816968655, 4.63410792546612480757153678877, 5.68432191750115603564306604043, 6.73418075659966203906616057614, 7.455099593311729107106343012136, 8.88931678447733712413099895761, 9.48028402858157521549545942103, 10.22786161698109542636114667099, 11.186735999806044343958370586517, 12.17510360016574163839095414730, 13.04385628192742684833553837374, 13.76886342156297184887053299722, 14.08635293240334613639569232293, 14.79288560105668446191535069542, 16.03500593897529781485446058014, 17.16181985230501856157662578620, 18.06231488950999325289632638843, 18.65722391983660119404867444013, 19.654931707806615042032539764782, 20.14437210408288880777574777563, 21.02631576425056106743813525539, 21.41207202351766447281693543297

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