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Properties

Label 1-167-167.36-r0-0-0
Degree $1$
Conductor $167$
Sign $-0.126 + 0.991i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (−0.881 − 0.472i)2-s + (0.0567 − 0.998i)3-s + (0.553 + 0.832i)4-s + (−0.752 − 0.658i)5-s + (−0.521 + 0.853i)6-s + (−0.999 − 0.0378i)7-s + (−0.0944 − 0.995i)8-s + (−0.993 − 0.113i)9-s + (0.351 + 0.936i)10-s + (0.280 + 0.959i)11-s + (0.862 − 0.505i)12-s + (−0.243 + 0.969i)13-s + (0.862 + 0.505i)14-s + (−0.700 + 0.713i)15-s + (−0.387 + 0.922i)16-s + (−0.914 − 0.404i)17-s + ⋯
L(s)  = 1  + (−0.881 − 0.472i)2-s + (0.0567 − 0.998i)3-s + (0.553 + 0.832i)4-s + (−0.752 − 0.658i)5-s + (−0.521 + 0.853i)6-s + (−0.999 − 0.0378i)7-s + (−0.0944 − 0.995i)8-s + (−0.993 − 0.113i)9-s + (0.351 + 0.936i)10-s + (0.280 + 0.959i)11-s + (0.862 − 0.505i)12-s + (−0.243 + 0.969i)13-s + (0.862 + 0.505i)14-s + (−0.700 + 0.713i)15-s + (−0.387 + 0.922i)16-s + (−0.914 − 0.404i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $-0.126 + 0.991i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ -0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0004117574104 + 0.0004674776556i\)
\(L(\frac12)\) \(\approx\) \(0.0004117574104 + 0.0004674776556i\)
\(L(1)\) \(\approx\) \(0.3624519511 - 0.2038647723i\)
\(L(1)\) \(\approx\) \(0.3624519511 - 0.2038647723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.881 - 0.472i)T \)
3 \( 1 + (0.0567 - 0.998i)T \)
5 \( 1 + (-0.752 - 0.658i)T \)
7 \( 1 + (-0.999 - 0.0378i)T \)
11 \( 1 + (0.280 + 0.959i)T \)
13 \( 1 + (-0.243 + 0.969i)T \)
17 \( 1 + (-0.914 - 0.404i)T \)
19 \( 1 + (-0.644 - 0.764i)T \)
23 \( 1 + (0.489 - 0.872i)T \)
29 \( 1 + (0.489 + 0.872i)T \)
31 \( 1 + (-0.316 + 0.948i)T \)
37 \( 1 + (-0.993 + 0.113i)T \)
41 \( 1 + (-0.800 - 0.599i)T \)
43 \( 1 + (-0.965 + 0.261i)T \)
47 \( 1 + (0.776 + 0.629i)T \)
53 \( 1 + (0.988 - 0.150i)T \)
59 \( 1 + (-0.914 + 0.404i)T \)
61 \( 1 + (-0.942 - 0.334i)T \)
67 \( 1 + (-0.752 + 0.658i)T \)
71 \( 1 + (-0.584 - 0.811i)T \)
73 \( 1 + (-0.387 - 0.922i)T \)
79 \( 1 + (0.929 - 0.369i)T \)
83 \( 1 + (-0.881 + 0.472i)T \)
89 \( 1 + (-0.700 - 0.713i)T \)
97 \( 1 + (-0.316 - 0.948i)T \)
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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

−27.17691728395274147330919446528, −26.58093059875050555464727323502, −25.76680515937763423299446981491, −24.80933342771373730348207745925, −23.38598602771342240026732582800, −22.64053687293158694415691886486, −21.63489474523454786198452702631, −20.151015310338692711550696767584, −19.50325898649230634923089689016, −18.70033884733016942873231535946, −17.25767697293415706542503005581, −16.45389729054425168956254850983, −15.393739076226022126369579829547, −15.119929228388983700830963166226, −13.66821763384242917335749272111, −11.79886149834260339860499737747, −10.75811707702839070365953564922, −10.10150521919731205958659894096, −8.90333338703302705957885022822, −7.99759524246565583729838120052, −6.59057945915278392767616390519, −5.64235808728334902884932984609, −3.85308622688293935760982758203, −2.78463873723465071041790675706, −0.00064827509856262624132686211, 1.65814032519778543863623537951, 2.968200465243907510355448711904, 4.488998140527618762056540841746, 6.81989337776788033961448597273, 7.05833420571771884985854809760, 8.6804445828908348682291310586, 9.14708917422011044597476243115, 10.74768399217083945021450884077, 12.031857597195408827325869966829, 12.45751605790500124605665602739, 13.47316471061627891456126917234, 15.2296298946889315412503320944, 16.3851742856045250192987136426, 17.16083626749517445115916165247, 18.24736669544974805764241132346, 19.314051751243624371535424010335, 19.74073470110129771094555249075, 20.55771491199055762467650509938, 22.09150536735968456348419673125, 23.17526070374513310091840650462, 24.205342737284076635668216187672, 25.15469169541356229682558352301, 25.96024018837801553977997641576, 26.944888250075364131656007147634, 28.225825774581730019743070618849

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