Properties

Degree $2$
Conductor $177$
Sign $0.932 - 0.362i$
Motivic weight $1$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.932 - 0.362i$
Motivic weight: \(1\)
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.932 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40949 + 0.451624i\)
\(L(\frac12)\) \(\approx\) \(2.40949 + 0.451624i\)
\(L(\frac{3}{2})\) 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

−12.79239121574143400178249295141, −12.01525560963580616197004378668, −11.18857388146672711544493949131, −9.867354043218677604597118083558, −8.922954947609475960660224501836, −7.15110472524474913082286996705, −5.73588014074355941509733657907, −5.05037778064249550871565612209, −4.12915229721933757523501965293, −2.88594219572959946677667570769, 2.66613238351236441487762617684, 3.30315047954845361611129105567, 5.34966373185887616258985106069, 6.07714254149424880034628171358, 7.05584505230003182144693412766, 7.88094010799618399086783381289, 10.19495458171691149034499601125, 11.03273330405804771969945625987, 12.03412814739059959558891034800, 12.88906610336179455177866604690

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