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Properties

Label 1-167-167.21-r0-0-0
Degree $1$
Conductor $167$
Sign $0.227 + 0.973i$
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  + (0.862 + 0.505i)2-s + (0.954 + 0.298i)3-s + (0.489 + 0.872i)4-s + (−0.169 + 0.985i)5-s + (0.672 + 0.739i)6-s + (−0.316 − 0.948i)7-s + (−0.0189 + 0.999i)8-s + (0.822 + 0.569i)9-s + (−0.644 + 0.764i)10-s + (0.0567 − 0.998i)11-s + (0.206 + 0.978i)12-s + (−0.965 − 0.261i)13-s + (0.206 − 0.978i)14-s + (−0.455 + 0.890i)15-s + (−0.521 + 0.853i)16-s + (−0.387 − 0.922i)17-s + ⋯
L(s)  = 1  + (0.862 + 0.505i)2-s + (0.954 + 0.298i)3-s + (0.489 + 0.872i)4-s + (−0.169 + 0.985i)5-s + (0.672 + 0.739i)6-s + (−0.316 − 0.948i)7-s + (−0.0189 + 0.999i)8-s + (0.822 + 0.569i)9-s + (−0.644 + 0.764i)10-s + (0.0567 − 0.998i)11-s + (0.206 + 0.978i)12-s + (−0.965 − 0.261i)13-s + (0.206 − 0.978i)14-s + (−0.455 + 0.890i)15-s + (−0.521 + 0.853i)16-s + (−0.387 − 0.922i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.227 + 0.973i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.227 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.750066688 + 1.388594497i\)
\(L(\frac12)\) \(\approx\) \(1.750066688 + 1.388594497i\)
\(L(1)\) \(\approx\) \(1.750585182 + 0.8998315234i\)
\(L(1)\) \(\approx\) \(1.750585182 + 0.8998315234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.862 + 0.505i)T \)
3 \( 1 + (0.954 + 0.298i)T \)
5 \( 1 + (-0.169 + 0.985i)T \)
7 \( 1 + (-0.316 - 0.948i)T \)
11 \( 1 + (0.0567 - 0.998i)T \)
13 \( 1 + (-0.965 - 0.261i)T \)
17 \( 1 + (-0.387 - 0.922i)T \)
19 \( 1 + (0.898 + 0.438i)T \)
23 \( 1 + (-0.914 + 0.404i)T \)
29 \( 1 + (-0.914 - 0.404i)T \)
31 \( 1 + (0.929 - 0.369i)T \)
37 \( 1 + (0.822 - 0.569i)T \)
41 \( 1 + (0.726 - 0.686i)T \)
43 \( 1 + (0.776 + 0.629i)T \)
47 \( 1 + (-0.881 - 0.472i)T \)
53 \( 1 + (0.280 + 0.959i)T \)
59 \( 1 + (-0.387 + 0.922i)T \)
61 \( 1 + (-0.243 + 0.969i)T \)
67 \( 1 + (-0.169 - 0.985i)T \)
71 \( 1 + (-0.982 + 0.188i)T \)
73 \( 1 + (-0.521 - 0.853i)T \)
79 \( 1 + (0.997 + 0.0756i)T \)
83 \( 1 + (0.862 - 0.505i)T \)
89 \( 1 + (-0.455 - 0.890i)T \)
97 \( 1 + (0.929 + 0.369i)T \)
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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

−27.8053627615028349934233935266, −26.265611982835684109932571328626, −25.13005459954014144283614937297, −24.47337373866481336698695217804, −23.81606556528531853524776275998, −22.372449019127183468926813076559, −21.53842997656624089944389398045, −20.48727464309141900376237773542, −19.8438806804401476751885784146, −19.10065439449770276093661278732, −17.80881343032186759692461051369, −16.095353489354068905717797392088, −15.23760179688686391611132069474, −14.45041752112257118534460787933, −13.140275021121057218656386019513, −12.52147312230252279972454787925, −11.80910860358450971378349354811, −9.85607539950411132707809453451, −9.23200417100520617850345077575, −7.858173340327512069379422053790, −6.50035956638786039609475790464, −5.05479605447187853656236000729, −4.07465705189904700905855348303, −2.62471671013983630463774591198, −1.67318571622248860970036990101, 2.56269924024057225223837262473, 3.40766907938723563972999543122, 4.35943504842987265619089910095, 5.98417889547292930203024058148, 7.38626057096886399091572348944, 7.735756636934289247613914092265, 9.47836555521919408260484729249, 10.64359027218457155333562302050, 11.79655979669846080881321709259, 13.39243447320718388133581141565, 13.94066486023339938916097801554, 14.709258937275971931143086640449, 15.77462119895549697615756995960, 16.54604024359220428797896953958, 17.97793864172686112845390072682, 19.347364395842502563807646652960, 20.11664633519095927150086704836, 21.17868152815601601331047877308, 22.224156085442551022778808032880, 22.77804182664888530570552443977, 24.15523301589091969155664124753, 24.82301891167128356900040487171, 26.08139293827395225352419159905, 26.56828795822109583032980923382, 27.19965802037068184156528571618

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