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Properties

Label 2-177-177.176-c1-0-7
Degree $2$
Conductor $177$
Sign $0.736 + 0.676i$
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  − 2.47·2-s + (−0.321 + 1.70i)3-s + 4.11·4-s − 2.50i·5-s + (0.793 − 4.20i)6-s − 2.11·7-s − 5.22·8-s + (−2.79 − 1.09i)9-s + 6.19i·10-s + 3.64·11-s + (−1.32 + 7.00i)12-s − 4.69i·13-s + 5.22·14-s + (4.26 + 0.804i)15-s + 4.70·16-s − 2.79i·17-s + ⋯
L(s)  = 1  − 1.74·2-s + (−0.185 + 0.982i)3-s + 2.05·4-s − 1.12i·5-s + (0.324 − 1.71i)6-s − 0.799·7-s − 1.84·8-s + (−0.931 − 0.364i)9-s + 1.96i·10-s + 1.09·11-s + (−0.381 + 2.02i)12-s − 1.30i·13-s + 1.39·14-s + (1.10 + 0.207i)15-s + 1.17·16-s − 0.677i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.736 + 0.676i$
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: \((2,\ 177,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401263 - 0.156427i\)
\(L(\frac12)\) \(\approx\) \(0.401263 - 0.156427i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.321 - 1.70i)T \)
59 \( 1 + (4.06 - 6.52i)T \)
good2 \( 1 + 2.47T + 2T^{2} \)
5 \( 1 + 2.50iT - 5T^{2} \)
7 \( 1 + 2.11T + 7T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + 4.69iT - 13T^{2} \)
17 \( 1 + 2.79iT - 17T^{2} \)
19 \( 1 - 6.70T + 19T^{2} \)
23 \( 1 - 0.885T + 23T^{2} \)
29 \( 1 + 1.50iT - 29T^{2} \)
31 \( 1 + 6.91iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 - 0.288iT - 41T^{2} \)
43 \( 1 + 7.70iT - 43T^{2} \)
47 \( 1 + 9.47T + 47T^{2} \)
53 \( 1 + 9.31iT - 53T^{2} \)
61 \( 1 + 9.92iT - 61T^{2} \)
67 \( 1 - 4.69iT - 67T^{2} \)
71 \( 1 - 2.69iT - 71T^{2} \)
73 \( 1 + 2.47iT - 73T^{2} \)
79 \( 1 + 3.56T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 5.93T + 89T^{2} \)
97 \( 1 - 1.67iT - 97T^{2} \)
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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

−12.04094012697313358453300292278, −11.37570959456421513559153114334, −10.01186759433293958587231127948, −9.614687645542346171344833874583, −8.852039603942354575159401261932, −7.891580137649507359295222743853, −6.43489512827673301310510265015, −5.10532225802669375944002241310, −3.25349456703417269689273796349, −0.75622189551127714428786083041, 1.56974727555100750332175828155, 3.11793327663736736822782404348, 6.27427348414893496140922450500, 6.82511365034063810918415631404, 7.53163570493608929433771979012, 8.897180111337582451536529196053, 9.606336239227786115555751622526, 10.81867691407162025313538154438, 11.49103243907091619882679733346, 12.38380191665108178527122364800

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