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Properties

Label 2-177-177.176-c1-0-3
Degree $2$
Conductor $177$
Sign $0.696 - 0.717i$
Analytic cond. $1.41335$
Root an. cond. $1.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 − 1.20i)3-s − 2·4-s + 3.58i·5-s + 5.15·7-s + (0.0927 + 2.99i)9-s + (2.48 + 2.41i)12-s + (4.32 − 4.45i)15-s + 4·16-s + 7.68i·17-s − 2.30·19-s − 7.17i·20-s + (−6.41 − 6.22i)21-s − 7.85·25-s + (3.5 − 3.84i)27-s − 10.3·28-s − 3.64i·29-s + ⋯
L(s)  = 1  + (−0.717 − 0.696i)3-s − 4-s + 1.60i·5-s + 1.95·7-s + (0.0309 + 0.999i)9-s + (0.717 + 0.696i)12-s + (1.11 − 1.15i)15-s + 16-s + 1.86i·17-s − 0.528·19-s − 1.60i·20-s + (−1.40 − 1.35i)21-s − 1.57·25-s + (0.673 − 0.739i)27-s − 1.95·28-s − 0.677i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.789212 + 0.334070i\)
\(L(\frac12)\) \(\approx\) \(0.789212 + 0.334070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 + 1.20i)T \)
59 \( 1 + 7.68iT \)
good2 \( 1 + 2T^{2} \)
5 \( 1 - 3.58iT - 5T^{2} \)
7 \( 1 - 5.15T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 7.68iT - 17T^{2} \)
19 \( 1 + 2.30T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 3.64iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 0.0624iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.4iT - 53T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 7.68iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.84049321681352322028510340244, −11.65298783091615526064075650452, −10.88827535187030217298960639791, −10.26641059138070064135458963944, −8.341750769084404328942501304358, −7.79910666977781083640403494494, −6.48410664301203497605651106638, −5.36430593724127608616428479917, −4.08726542167024137495585613196, −1.90759366497427453963034178453, 1.01085268182840106376343442750, 4.32469803135168192830013387671, 4.87970349241540486350579246227, 5.45614113122990597672285857486, 7.73243397171610173664706880439, 8.804254986455005981271907048541, 9.274662667136823741138594174031, 10.62958286648556421964805883303, 11.72680220167313986597668436563, 12.33815347320516769639360902408

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