Properties

Label 2-177-177.101-c1-0-4
Degree $2$
Conductor $177$
Sign $0.206 - 0.978i$
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  + (−0.369 + 0.544i)2-s + (0.293 − 1.70i)3-s + (0.580 + 1.45i)4-s + (−1.46 + 3.16i)5-s + (0.821 + 0.790i)6-s + (0.554 + 1.99i)7-s + (−2.29 − 0.504i)8-s + (−2.82 − 1.00i)9-s + (−1.18 − 1.96i)10-s + (2.57 + 2.44i)11-s + (2.65 − 0.563i)12-s + (0.370 − 3.40i)13-s + (−1.29 − 0.435i)14-s + (4.97 + 3.43i)15-s + (−1.15 + 1.09i)16-s + (2.77 + 0.769i)17-s + ⋯
L(s)  = 1  + (−0.261 + 0.385i)2-s + (0.169 − 0.985i)3-s + (0.290 + 0.727i)4-s + (−0.655 + 1.41i)5-s + (0.335 + 0.322i)6-s + (0.209 + 0.754i)7-s + (−0.810 − 0.178i)8-s + (−0.942 − 0.333i)9-s + (−0.374 − 0.622i)10-s + (0.777 + 0.736i)11-s + (0.766 − 0.162i)12-s + (0.102 − 0.945i)13-s + (−0.345 − 0.116i)14-s + (1.28 + 0.885i)15-s + (−0.288 + 0.273i)16-s + (0.672 + 0.186i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783383 + 0.635589i\)
\(L(\frac12)\) \(\approx\) \(0.783383 + 0.635589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.293 + 1.70i)T \)
59 \( 1 + (4.86 + 5.94i)T \)
good2 \( 1 + (0.369 - 0.544i)T + (-0.740 - 1.85i)T^{2} \)
5 \( 1 + (1.46 - 3.16i)T + (-3.23 - 3.81i)T^{2} \)
7 \( 1 + (-0.554 - 1.99i)T + (-5.99 + 3.60i)T^{2} \)
11 \( 1 + (-2.57 - 2.44i)T + (0.595 + 10.9i)T^{2} \)
13 \( 1 + (-0.370 + 3.40i)T + (-12.6 - 2.79i)T^{2} \)
17 \( 1 + (-2.77 - 0.769i)T + (14.5 + 8.76i)T^{2} \)
19 \( 1 + (-5.15 - 3.91i)T + (5.08 + 18.3i)T^{2} \)
23 \( 1 + (3.83 + 7.22i)T + (-12.9 + 19.0i)T^{2} \)
29 \( 1 + (-4.32 + 2.92i)T + (10.7 - 26.9i)T^{2} \)
31 \( 1 + (2.30 + 3.03i)T + (-8.29 + 29.8i)T^{2} \)
37 \( 1 + (-1.23 - 5.60i)T + (-33.5 + 15.5i)T^{2} \)
41 \( 1 + (-1.75 - 0.929i)T + (23.0 + 33.9i)T^{2} \)
43 \( 1 + (-1.47 - 1.55i)T + (-2.32 + 42.9i)T^{2} \)
47 \( 1 + (1.65 - 0.764i)T + (30.4 - 35.8i)T^{2} \)
53 \( 1 + (-3.76 + 6.24i)T + (-24.8 - 46.8i)T^{2} \)
61 \( 1 + (5.90 + 4.00i)T + (22.5 + 56.6i)T^{2} \)
67 \( 1 + (-2.37 + 10.7i)T + (-60.8 - 28.1i)T^{2} \)
71 \( 1 + (-6.58 - 14.2i)T + (-45.9 + 54.1i)T^{2} \)
73 \( 1 + (0.980 - 2.91i)T + (-58.1 - 44.1i)T^{2} \)
79 \( 1 + (0.440 - 8.12i)T + (-78.5 - 8.54i)T^{2} \)
83 \( 1 + (0.0614 - 0.374i)T + (-78.6 - 26.5i)T^{2} \)
89 \( 1 + (-6.86 - 10.1i)T + (-32.9 + 82.6i)T^{2} \)
97 \( 1 + (-0.271 - 0.804i)T + (-77.2 + 58.7i)T^{2} \)
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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

−12.49768233733500963572704447313, −12.08754467890818779006968319632, −11.27253691193069378493408144018, −9.842598506706790629867132945341, −8.259883050419853302916636663660, −7.80724976418656800669329406642, −6.82427441663621845821295498973, −6.02438484459075664135931039376, −3.54251348302674397654502719320, −2.52293873309491453672113486173, 1.09979730051243445655149436823, 3.57742337212135551877610446852, 4.70556852879630197071169295624, 5.72852538483585546516273223880, 7.52060494689950988057184229616, 9.007711330564266650006805080356, 9.239772492187313492170634921154, 10.49004706705685782240237353975, 11.54560318458504625107870222793, 11.92241665014583748147483501745

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