Properties

Label 2-177-177.101-c3-0-14
Degree $2$
Conductor $177$
Sign $-0.792 - 0.610i$
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  + (0.428 − 0.632i)2-s + (1.93 + 4.82i)3-s + (2.74 + 6.88i)4-s + (−6.52 + 14.1i)5-s + (3.88 + 0.845i)6-s + (−5.03 − 18.1i)7-s + (11.5 + 2.53i)8-s + (−19.5 + 18.6i)9-s + (6.12 + 10.1i)10-s + (33.9 + 32.1i)11-s + (−27.9 + 26.5i)12-s + (3.16 − 29.0i)13-s + (−13.6 − 4.58i)14-s + (−80.6 − 4.19i)15-s + (−36.5 + 34.6i)16-s + (18.6 + 5.18i)17-s + ⋯
L(s)  = 1  + (0.151 − 0.223i)2-s + (0.372 + 0.928i)3-s + (0.343 + 0.861i)4-s + (−0.583 + 1.26i)5-s + (0.264 + 0.0575i)6-s + (−0.271 − 0.978i)7-s + (0.508 + 0.111i)8-s + (−0.723 + 0.690i)9-s + (0.193 + 0.321i)10-s + (0.929 + 0.880i)11-s + (−0.671 + 0.638i)12-s + (0.0674 − 0.620i)13-s + (−0.259 − 0.0876i)14-s + (−1.38 − 0.0722i)15-s + (−0.570 + 0.540i)16-s + (0.266 + 0.0740i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.792 - 0.610i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ -0.792 - 0.610i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.568497 + 1.66884i\)
\(L(\frac12)\) \(\approx\) \(0.568497 + 1.66884i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.93 - 4.82i)T \)
59 \( 1 + (430. + 140. i)T \)
good2 \( 1 + (-0.428 + 0.632i)T + (-2.96 - 7.43i)T^{2} \)
5 \( 1 + (6.52 - 14.1i)T + (-80.9 - 95.2i)T^{2} \)
7 \( 1 + (5.03 + 18.1i)T + (-293. + 176. i)T^{2} \)
11 \( 1 + (-33.9 - 32.1i)T + (72.0 + 1.32e3i)T^{2} \)
13 \( 1 + (-3.16 + 29.0i)T + (-2.14e3 - 472. i)T^{2} \)
17 \( 1 + (-18.6 - 5.18i)T + (4.20e3 + 2.53e3i)T^{2} \)
19 \( 1 + (91.9 + 69.8i)T + (1.83e3 + 6.60e3i)T^{2} \)
23 \( 1 + (16.9 + 32.0i)T + (-6.82e3 + 1.00e4i)T^{2} \)
29 \( 1 + (97.0 - 65.8i)T + (9.02e3 - 2.26e4i)T^{2} \)
31 \( 1 + (-110. - 144. i)T + (-7.96e3 + 2.87e4i)T^{2} \)
37 \( 1 + (-36.4 - 165. i)T + (-4.59e4 + 2.12e4i)T^{2} \)
41 \( 1 + (-411. - 218. i)T + (3.86e4 + 5.70e4i)T^{2} \)
43 \( 1 + (48.6 + 51.3i)T + (-4.30e3 + 7.93e4i)T^{2} \)
47 \( 1 + (-112. + 51.8i)T + (6.72e4 - 7.91e4i)T^{2} \)
53 \( 1 + (-125. + 208. i)T + (-6.97e4 - 1.31e5i)T^{2} \)
61 \( 1 + (-718. - 487. i)T + (8.40e4 + 2.10e5i)T^{2} \)
67 \( 1 + (105. - 478. i)T + (-2.72e5 - 1.26e5i)T^{2} \)
71 \( 1 + (-277. - 598. i)T + (-2.31e5 + 2.72e5i)T^{2} \)
73 \( 1 + (-339. + 1.00e3i)T + (-3.09e5 - 2.35e5i)T^{2} \)
79 \( 1 + (42.2 - 778. i)T + (-4.90e5 - 5.33e4i)T^{2} \)
83 \( 1 + (66.9 - 408. i)T + (-5.41e5 - 1.82e5i)T^{2} \)
89 \( 1 + (-191. - 282. i)T + (-2.60e5 + 6.54e5i)T^{2} \)
97 \( 1 + (316. + 938. i)T + (-7.26e5 + 5.52e5i)T^{2} \)
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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

−12.54679951748154809929171868328, −11.34132670032595799456126396469, −10.75446225830289602091880545741, −9.916842130921119648303390142540, −8.486987609582186158847365247386, −7.39148850786661267237765608671, −6.66571090494896631123163446326, −4.39479689253063750721964294649, −3.67248801622595345243785586316, −2.66063844202688187618877208407, 0.72986880854463378612394551536, 2.04539726546330867260408748701, 4.06354404947790535103827454817, 5.71534593092266952730382084455, 6.30301292249992955114809675351, 7.76193996278857128905439303918, 8.799705152637767725533986448399, 9.386278132275049007272435220635, 11.22432656558309844684759102737, 12.01842960283966207261727960838

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