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Properties

Label 2-177-177.176-c5-0-59
Degree $2$
Conductor $177$
Sign $-0.332 + 0.943i$
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  + (−14.7 − 5.18i)3-s − 32·4-s − 61.6i·5-s + 145.·7-s + (189. + 152. i)9-s + (470. + 165. i)12-s + (−319. + 906. i)15-s + 1.02e3·16-s + 683. i·17-s + 2.16e3·19-s + 1.97e3i·20-s + (−2.13e3 − 754. i)21-s − 679.·25-s + (−1.99e3 − 3.22e3i)27-s − 4.65e3·28-s − 5.70e3i·29-s + ⋯
L(s)  = 1  + (−0.943 − 0.332i)3-s − 4-s − 1.10i·5-s + 1.12·7-s + (0.778 + 0.627i)9-s + (0.943 + 0.332i)12-s + (−0.367 + 1.04i)15-s + 16-s + 0.573i·17-s + 1.37·19-s + 1.10i·20-s + (−1.05 − 0.373i)21-s − 0.217·25-s + (−0.525 − 0.850i)27-s − 1.12·28-s − 1.25i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.332 + 0.943i$
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: \((2,\ 177,\ (\ :5/2),\ -0.332 + 0.943i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.105145556\)
\(L(\frac12)\) \(\approx\) \(1.105145556\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (14.7 + 5.18i)T \)
59 \( 1 - 2.67e4iT \)
good2 \( 1 + 32T^{2} \)
5 \( 1 + 61.6iT - 3.12e3T^{2} \)
7 \( 1 - 145.T + 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 - 683. iT - 1.41e6T^{2} \)
19 \( 1 - 2.16e3T + 2.47e6T^{2} \)
23 \( 1 + 6.43e6T^{2} \)
29 \( 1 + 5.70e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 + 1.83e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.47e8T^{2} \)
47 \( 1 + 2.29e8T^{2} \)
53 \( 1 - 8.35e3iT - 4.18e8T^{2} \)
61 \( 1 - 8.44e8T^{2} \)
67 \( 1 - 1.35e9T^{2} \)
71 \( 1 + 5.94e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 + 1.55e3T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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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

−11.73997553637135285736168915765, −10.55994238205281833638874061025, −9.424295318696453234877581049103, −8.379896163922879331173722680899, −7.55104593764214280159974916458, −5.73533240695052914924884328931, −5.02722651034342416519027942209, −4.19166889560640835876225326701, −1.50399233522670350682208692355, −0.53778461490728733386474666220, 1.12485488483231663446804191111, 3.31704531904092638713121883553, 4.69371959999365124516558447782, 5.43893393923653429228546132682, 6.84422700201162113832467607545, 7.913742721162134966829278932034, 9.310506208010546241398554059169, 10.21582722668111788408469155808, 11.13278095846932941562438845674, 11.81689678836557939717594120177

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