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Properties

Label 1-164-164.71-r0-0-0
Degree $1$
Conductor $164$
Sign $0.976 + 0.217i$
Analytic cond. $0.761612$
Root an. cond. $0.761612$
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  + (0.707 + 0.707i)3-s + (−0.587 − 0.809i)5-s + (0.891 − 0.453i)7-s + i·9-s + (0.156 + 0.987i)11-s + (0.453 − 0.891i)13-s + (0.156 − 0.987i)15-s + (0.987 − 0.156i)17-s + (−0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (0.309 + 0.951i)23-s + (−0.309 + 0.951i)25-s + (−0.707 + 0.707i)27-s + (0.987 + 0.156i)29-s + (−0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.587 − 0.809i)5-s + (0.891 − 0.453i)7-s + i·9-s + (0.156 + 0.987i)11-s + (0.453 − 0.891i)13-s + (0.156 − 0.987i)15-s + (0.987 − 0.156i)17-s + (−0.453 − 0.891i)19-s + (0.951 + 0.309i)21-s + (0.309 + 0.951i)23-s + (−0.309 + 0.951i)25-s + (−0.707 + 0.707i)27-s + (0.987 + 0.156i)29-s + (−0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(164\)    =    \(2^{2} \cdot 41\)
Sign: $0.976 + 0.217i$
Analytic conductor: \(0.761612\)
Root analytic conductor: \(0.761612\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{164} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 164,\ (0:\ ),\ 0.976 + 0.217i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.404214891 + 0.1544325504i\)
\(L(\frac12)\) \(\approx\) \(1.404214891 + 0.1544325504i\)
\(L(1)\) \(\approx\) \(1.290877643 + 0.1129662836i\)
\(L(1)\) \(\approx\) \(1.290877643 + 0.1129662836i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.891 - 0.453i)T \)
11 \( 1 + (0.156 + 0.987i)T \)
13 \( 1 + (0.453 - 0.891i)T \)
17 \( 1 + (0.987 - 0.156i)T \)
19 \( 1 + (-0.453 - 0.891i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.951 - 0.309i)T \)
67 \( 1 + (-0.156 + 0.987i)T \)
71 \( 1 + (-0.156 - 0.987i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.891 + 0.453i)T \)
97 \( 1 + (-0.156 + 0.987i)T \)
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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

−27.37216728888134027838944159849, −26.74724811509163097813540197156, −25.70726941270732341536365160472, −24.77861552831474631309943185789, −23.82520108444172664690813157784, −23.14219498327650112758547162849, −21.61748703771780593981929809585, −20.91860413398451807090810590316, −19.58728234620703046178474764394, −18.684250461916587121886167919050, −18.38150446540361839469638083359, −16.82239403620963723854897575059, −15.51052583471570895240198890369, −14.2891983987559821544148848898, −14.16979387980176527442725975570, −12.42196577672194830640838450414, −11.62085997574546576600854552390, −10.51788846271359516539873339940, −8.771688487384753149928506583432, −8.16517727896727587622201868822, −7.00884706983303207527980409492, −5.918991953980159111455840715847, −4.01856289733747394828946347070, −2.923850966162893927247387604959, −1.55251064808568464754997734885, 1.515997404832387756378498957260, 3.3046902492822516517258196862, 4.474850539993520790923673975724, 5.20808814721347786256612620147, 7.43640154773539221098929212590, 8.16264343273913688515690869696, 9.21231820100949429454140581622, 10.35535764032974932865311358420, 11.44989366625256635685235071637, 12.72793659826122249639357250168, 13.81439596718484885943314065929, 14.99520673835121597142625832245, 15.59368167793231619052308804649, 16.80296876570954565354992894902, 17.6945308706830546562279839253, 19.23869970805520124225056651992, 20.25418877559178905069270883918, 20.61999984228882412784612582909, 21.65222265401114340141013530740, 23.03640901408415241427500569451, 23.82673615162072444039920673005, 25.06380188751981718170965611813, 25.69746397151626346830983551092, 27.04246714297047466487004288080, 27.68718059956273804386262476514

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