Properties

Label 2-177-177.176-c3-0-9
Degree $2$
Conductor $177$
Sign $-0.952 + 0.303i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.57 + 4.95i)3-s − 8·4-s + 14.3i·5-s + 5.45·7-s + (−22.0 + 15.6i)9-s + (−12.6 − 39.6i)12-s + (−71.0 + 22.6i)15-s + 64·16-s − 61.4i·17-s − 159.·19-s − 114. i·20-s + (8.59 + 27.0i)21-s − 80.8·25-s + (−111. − 84.4i)27-s − 43.6·28-s − 3.27i·29-s + ⋯
L(s)  = 1  + (0.303 + 0.952i)3-s − 4-s + 1.28i·5-s + 0.294·7-s + (−0.816 + 0.578i)9-s + (−0.303 − 0.952i)12-s + (−1.22 + 0.389i)15-s + 16-s − 0.876i·17-s − 1.92·19-s − 1.28i·20-s + (0.0893 + 0.280i)21-s − 0.646·25-s + (−0.798 − 0.602i)27-s − 0.294·28-s − 0.0209i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.952 + 0.303i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.952 + 0.303i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.117867 - 0.758907i\)
\(L(\frac12)\) \(\approx\) \(0.117867 - 0.758907i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.57 - 4.95i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 8T^{2} \)
5 \( 1 - 14.3iT - 125T^{2} \)
7 \( 1 - 5.45T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 61.4iT - 4.91e3T^{2} \)
19 \( 1 + 159.T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 3.27iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 - 450. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 66.3iT - 1.48e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 1.18e3iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 1.39e3T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−12.99709166017941076698722672525, −11.47094680310679674245537816486, −10.61301895567652623012931867625, −9.868583732353880663384382388751, −8.850156423830596864415932079742, −7.88568581792547979547884420356, −6.38726489432904388152097064027, −4.96808870443704163107983524371, −3.94397026756928190544901243350, −2.70309303190026504760401059898, 0.34200826286994347660510959214, 1.76306720346853333465005933389, 3.93232725477115576020955822793, 5.08288369691708611030722406455, 6.29291062911492051506644527401, 7.941712549700444723198189741399, 8.567013926483412586395556568972, 9.221034150226689550101459524693, 10.69102150833420392225325246239, 12.23481228606290600691422484774

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