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Properties

Label 1-176-176.109-r1-0-0
Degree $1$
Conductor $176$
Sign $0.382 - 0.923i$
Analytic cond. $18.9138$
Root an. cond. $18.9138$
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  + i·3-s − i·5-s + 7-s − 9-s − i·13-s + 15-s − 17-s − i·19-s + i·21-s − 23-s − 25-s − i·27-s − i·29-s + 31-s − i·35-s + ⋯
L(s)  = 1  + i·3-s − i·5-s + 7-s − 9-s − i·13-s + 15-s − 17-s − i·19-s + i·21-s − 23-s − 25-s − i·27-s − i·29-s + 31-s − i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(18.9138\)
Root analytic conductor: \(18.9138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 176,\ (1:\ ),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217523517 - 0.8135232053i\)
\(L(\frac12)\) \(\approx\) \(1.217523517 - 0.8135232053i\)
\(L(1)\) \(\approx\) \(1.056366367 - 0.07507364626i\)
\(L(1)\) \(\approx\) \(1.056366367 - 0.07507364626i\)

Euler product

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

−27.22968942660414115729994783182, −26.33693569192505190467280818231, −25.42256403133518068804178301528, −24.32334351726606675954247795674, −23.71131401777232036423777977628, −22.646823639612722481374863211031, −21.69455251194536322596528825725, −20.48530134545834795026896928225, −19.38046962322763050637370766181, −18.45924131246876233764709816871, −17.90106380308966185038551593157, −16.843103688839340241671131656422, −15.29820693066472233070884551792, −14.19477870483396625882774127437, −13.79106222914530393400653079574, −12.18603348329401714256351127800, −11.45121679646350086436957707022, −10.46120836549068679673638338778, −8.80933071766497236555099621675, −7.75417073487713468274832573856, −6.835598025160563912495796957076, −5.84595516720271032754102181921, −4.19031379651977528612157180356, −2.54461111676722973148478448659, −1.5587924267990017613404537521, 0.52247879121056840469058141227, 2.37659865041079952483394475111, 4.134404586787106190500530615159, 4.86487910309167195594222546997, 5.889041351414918402658409644, 7.8984029690414663506264673511, 8.67954103752612478555110086100, 9.71676750941607408837526025195, 10.91018531061433117169271872729, 11.77207053607899858068282222687, 13.10524335436337614496817855814, 14.21447882830397181705375791744, 15.41348484487966056268951714364, 15.96823407044303219147567502302, 17.38295478635281272152432999203, 17.684652861517098439555024852635, 19.663321421794508976816913260282, 20.35468232305212036269280978245, 21.13280314719309977820479636040, 21.988755811734829319788093937879, 23.10942006577023943215520364778, 24.267955817752754689161505697658, 24.946039516314663598747103305014, 26.26532617239547812702213495420, 27.08270629009934127470144627879

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