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Properties

Label 2-177-177.176-c5-0-42
Degree $2$
Conductor $177$
Sign $0.999 - 0.0173i$
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  − 9.95·2-s + (−15.5 + 1.06i)3-s + 67.0·4-s + 97.9i·5-s + (154. − 10.5i)6-s − 155.·7-s − 348.·8-s + (240. − 33.0i)9-s − 974. i·10-s + 637.·11-s + (−1.04e3 + 71.2i)12-s + 651. i·13-s + 1.55e3·14-s + (−103. − 1.52e3i)15-s + 1.32e3·16-s − 1.94e3i·17-s + ⋯
L(s)  = 1  − 1.75·2-s + (−0.997 + 0.0681i)3-s + 2.09·4-s + 1.75i·5-s + (1.75 − 0.119i)6-s − 1.20·7-s − 1.92·8-s + (0.990 − 0.135i)9-s − 3.08i·10-s + 1.58·11-s + (−2.09 + 0.142i)12-s + 1.06i·13-s + 2.11·14-s + (−0.119 − 1.74i)15-s + 1.29·16-s − 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.999 - 0.0173i$
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: \((2,\ 177,\ (\ :5/2),\ 0.999 - 0.0173i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3476116087\)
\(L(\frac12)\) \(\approx\) \(0.3476116087\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (15.5 - 1.06i)T \)
59 \( 1 + (-2.67e4 - 1.35e3i)T \)
good2 \( 1 + 9.95T + 32T^{2} \)
5 \( 1 - 97.9iT - 3.12e3T^{2} \)
7 \( 1 + 155.T + 1.68e4T^{2} \)
11 \( 1 - 637.T + 1.61e5T^{2} \)
13 \( 1 - 651. iT - 3.71e5T^{2} \)
17 \( 1 + 1.94e3iT - 1.41e6T^{2} \)
19 \( 1 + 604.T + 2.47e6T^{2} \)
23 \( 1 + 1.68e3T + 6.43e6T^{2} \)
29 \( 1 - 6.48e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.95e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.36e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.84e3iT - 1.15e8T^{2} \)
43 \( 1 + 8.73e3iT - 1.47e8T^{2} \)
47 \( 1 - 3.95e3T + 2.29e8T^{2} \)
53 \( 1 + 1.46e4iT - 4.18e8T^{2} \)
61 \( 1 + 2.82e4iT - 8.44e8T^{2} \)
67 \( 1 + 4.76e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.39e3iT - 1.80e9T^{2} \)
73 \( 1 + 1.60e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.04e4T + 3.07e9T^{2} \)
83 \( 1 - 5.88e4T + 3.93e9T^{2} \)
89 \( 1 - 7.53e4T + 5.58e9T^{2} \)
97 \( 1 - 1.00e5iT - 8.58e9T^{2} \)
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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

−11.43758182210342922799013483884, −10.68826323287585030501805418201, −9.657994632792394132670804200715, −9.337051009035772200387419837143, −7.30801399696350567090331189103, −6.73061272683708988180602854736, −6.30268021532585195553892319830, −3.67141827648483032980618003075, −2.11206753798959609422553645510, −0.37402317856696737258839906382, 0.73110419095678264918967624702, 1.50359034503664244034417481868, 4.09059871006691898632035658058, 5.86735539197278063860954518469, 6.57286328350997811822461975977, 8.035254251771883653214706625303, 8.862540865991857161143431198585, 9.746992686986097059581252854093, 10.40263223887368750061854100840, 11.75838839163327011898349018424

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