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Properties

Label 1-167-167.7-r0-0-0
Degree $1$
Conductor $167$
Sign $-0.115 - 0.993i$
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  + (−0.914 + 0.404i)2-s + (0.421 − 0.906i)3-s + (0.672 − 0.739i)4-s + (−0.243 − 0.969i)5-s + (−0.0189 + 0.999i)6-s + (0.726 − 0.686i)7-s + (−0.316 + 0.948i)8-s + (−0.644 − 0.764i)9-s + (0.614 + 0.788i)10-s + (0.822 − 0.569i)11-s + (−0.387 − 0.922i)12-s + (0.206 + 0.978i)13-s + (−0.387 + 0.922i)14-s + (−0.982 − 0.188i)15-s + (−0.0944 − 0.995i)16-s + (−0.455 − 0.890i)17-s + ⋯
L(s)  = 1  + (−0.914 + 0.404i)2-s + (0.421 − 0.906i)3-s + (0.672 − 0.739i)4-s + (−0.243 − 0.969i)5-s + (−0.0189 + 0.999i)6-s + (0.726 − 0.686i)7-s + (−0.316 + 0.948i)8-s + (−0.644 − 0.764i)9-s + (0.614 + 0.788i)10-s + (0.822 − 0.569i)11-s + (−0.387 − 0.922i)12-s + (0.206 + 0.978i)13-s + (−0.387 + 0.922i)14-s + (−0.982 − 0.188i)15-s + (−0.0944 − 0.995i)16-s + (−0.455 − 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ -0.115 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5682334749 - 0.6382123062i\)
\(L(\frac12)\) \(\approx\) \(0.5682334749 - 0.6382123062i\)
\(L(1)\) \(\approx\) \(0.7527707498 - 0.3569680763i\)
\(L(1)\) \(\approx\) \(0.7527707498 - 0.3569680763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.914 + 0.404i)T \)
3 \( 1 + (0.421 - 0.906i)T \)
5 \( 1 + (-0.243 - 0.969i)T \)
7 \( 1 + (0.726 - 0.686i)T \)
11 \( 1 + (0.822 - 0.569i)T \)
13 \( 1 + (0.206 + 0.978i)T \)
17 \( 1 + (-0.455 - 0.890i)T \)
19 \( 1 + (0.132 + 0.991i)T \)
23 \( 1 + (-0.700 + 0.713i)T \)
29 \( 1 + (-0.700 - 0.713i)T \)
31 \( 1 + (0.988 - 0.150i)T \)
37 \( 1 + (-0.644 + 0.764i)T \)
41 \( 1 + (0.954 - 0.298i)T \)
43 \( 1 + (0.553 - 0.832i)T \)
47 \( 1 + (0.489 - 0.872i)T \)
53 \( 1 + (-0.993 + 0.113i)T \)
59 \( 1 + (-0.455 + 0.890i)T \)
61 \( 1 + (0.862 - 0.505i)T \)
67 \( 1 + (-0.243 + 0.969i)T \)
71 \( 1 + (0.997 - 0.0756i)T \)
73 \( 1 + (-0.0944 + 0.995i)T \)
79 \( 1 + (0.280 + 0.959i)T \)
83 \( 1 + (-0.914 - 0.404i)T \)
89 \( 1 + (-0.982 + 0.188i)T \)
97 \( 1 + (0.988 + 0.150i)T \)
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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

−27.95388655074996578250400184638, −26.969268892693178247193658472242, −26.18119839357087233233790902736, −25.421401185941179246367429710503, −24.41744705138801388477329746440, −22.504605836695647274224456619655, −21.98543126539861681895940505424, −20.97122066443049511213691121803, −19.968356406337878306014053009724, −19.26472823908967448829761908223, −17.97228962586082597913152250254, −17.389941449766939640071992284947, −15.84050294531473573676147014162, −15.18905396048949996596002170355, −14.3271676756774428644945524147, −12.51154629672374433809297925602, −11.2332843740057131383595236915, −10.70229611373330402486174566900, −9.55836989766486105129186861258, −8.56612650276349410636870177279, −7.64319453230226627084190822320, −6.18345480967532332730145193205, −4.35571098400630656372020944500, −3.12096224435342250163065122272, −2.08423331145819810589144115729, 0.96610867170920377320325234587, 1.89923550960500530000357063270, 4.03163892403608549776412901091, 5.70667436039405596221989251471, 6.93808890252193209560157061890, 7.90452562796424565784339290111, 8.69091998243030778491968321034, 9.611430473246447448882601072325, 11.41179877878942073793265252770, 11.93483914609547954473997578192, 13.71040492598662949959424278718, 14.21408244202737493608918096625, 15.69377364326862389457745137473, 16.83524033399225485627718348636, 17.39573971819625815874093056723, 18.60825987913919568391287021213, 19.4209155740481257810938827733, 20.32295303054605933803528720365, 20.95661764659787515958287478714, 23.05143769125771441520722065454, 24.117274319765221347850344179024, 24.38455921828308001876793368331, 25.30387062731152340255960117862, 26.49068509958254276157308184118, 27.24614735681096723764387234836

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