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Properties

Label 2-177-177.176-c1-0-5
Degree $2$
Conductor $177$
Sign $0.273 - 0.961i$
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  + (1.66 + 0.474i)3-s − 2·4-s + 4.10i·5-s − 1.56·7-s + (2.55 + 1.57i)9-s + (−3.33 − 0.948i)12-s + (−1.94 + 6.84i)15-s + 4·16-s − 7.68i·17-s + 8.43·19-s − 8.21i·20-s + (−2.60 − 0.740i)21-s − 11.8·25-s + (3.50 + 3.84i)27-s + 3.12·28-s + 6.95i·29-s + ⋯
L(s)  = 1  + (0.961 + 0.273i)3-s − 4-s + 1.83i·5-s − 0.590·7-s + (0.850 + 0.526i)9-s + (−0.961 − 0.273i)12-s + (−0.502 + 1.76i)15-s + 16-s − 1.86i·17-s + 1.93·19-s − 1.83i·20-s + (−0.568 − 0.161i)21-s − 2.37·25-s + (0.673 + 0.739i)27-s + 0.590·28-s + 1.29i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.273 - 0.961i$
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: \((2,\ 177,\ (\ :1/2),\ 0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976756 + 0.737570i\)
\(L(\frac12)\) \(\approx\) \(0.976756 + 0.737570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 - 0.474i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 + 2T^{2} \)
5 \( 1 - 4.10iT - 5T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.68iT - 17T^{2} \)
19 \( 1 - 8.43T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6.95iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 5.37iT - 53T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 7.68iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 3.74T + 79T^{2} \)
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

−13.35275353853416767277614827116, −11.89237541795715698651765447974, −10.60432784961433455169619615127, −9.753477688576576823140838610031, −9.166644548740234108960814724897, −7.61802628858760819037488479545, −6.97986019234553109497116988072, −5.23223248435632000781719646647, −3.55441166804018318099222003724, −2.90372236907266968934588291955, 1.23454570450566170948994982900, 3.61760468280945731292185628031, 4.63005458357634827071880962273, 5.90702523545162679754096326291, 7.891056847060035876497254075009, 8.416783953346164491357024358962, 9.439352475781020408947679176767, 9.824790607351474518390977732396, 12.00769722351110063017256122665, 12.87561726059081000344376921224

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