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Properties

Label 2-177-177.176-c5-0-77
Degree $2$
Conductor $177$
Sign $-0.610 + 0.791i$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.63·2-s + (15.3 − 2.74i)3-s + 60.8·4-s − 48.0i·5-s + (−147. + 26.4i)6-s + 55.2·7-s − 278.·8-s + (227. − 84.1i)9-s + 463. i·10-s − 33.0·11-s + (934. − 166. i)12-s − 847. i·13-s − 532.·14-s + (−131. − 738. i)15-s + 733.·16-s − 1.12e3i·17-s + ⋯
L(s)  = 1  − 1.70·2-s + (0.984 − 0.175i)3-s + 1.90·4-s − 0.860i·5-s + (−1.67 + 0.299i)6-s + 0.426·7-s − 1.53·8-s + (0.938 − 0.346i)9-s + 1.46i·10-s − 0.0822·11-s + (1.87 − 0.334i)12-s − 1.39i·13-s − 0.726·14-s + (−0.151 − 0.847i)15-s + 0.716·16-s − 0.947i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.610 + 0.791i$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -0.610 + 0.791i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.053148883\)
\(L(\frac12)\) \(\approx\) \(1.053148883\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-15.3 + 2.74i)T \)
59 \( 1 + (-1.98e4 + 1.79e4i)T \)
good2 \( 1 + 9.63T + 32T^{2} \)
5 \( 1 + 48.0iT - 3.12e3T^{2} \)
7 \( 1 - 55.2T + 1.68e4T^{2} \)
11 \( 1 + 33.0T + 1.61e5T^{2} \)
13 \( 1 + 847. iT - 3.71e5T^{2} \)
17 \( 1 + 1.12e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.38e3T + 2.47e6T^{2} \)
23 \( 1 - 468.T + 6.43e6T^{2} \)
29 \( 1 - 408. iT - 2.05e7T^{2} \)
31 \( 1 - 5.08e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.57e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.69e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.11e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.20e4T + 2.29e8T^{2} \)
53 \( 1 - 9.98e3iT - 4.18e8T^{2} \)
61 \( 1 + 5.04e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.40e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.89e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.36e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.19e4T + 3.07e9T^{2} \)
83 \( 1 + 3.15e4T + 3.93e9T^{2} \)
89 \( 1 - 5.23e4T + 5.58e9T^{2} \)
97 \( 1 - 2.50e4iT - 8.58e9T^{2} \)
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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

−11.05545353862231846235134706705, −10.07885635287565291451993245379, −9.217990132139836337375931533268, −8.377669958441402230646018771089, −7.905984407205892034553438599046, −6.76428993208376641463793726877, −4.90013397890523030530581895154, −2.94389101267618124936122250004, −1.59590029687083066169891730548, −0.52427292336542579293275765615, 1.59787117796514017933687308803, 2.51608794195458158303961337937, 4.14619406342332140357351691721, 6.51468995900207134484807016914, 7.32054653440315918777609181878, 8.325624740704434795398159008068, 9.007001522820624806198485839104, 10.01788306948471100513156684880, 10.73730085596997742672183257301, 11.61753667092150694027289818630

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