Properties

Label 2-177-177.101-c3-0-23
Degree $2$
Conductor $177$
Sign $-0.138 - 0.990i$
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  + (0.171 − 0.252i)2-s + (4.85 + 1.85i)3-s + (2.92 + 7.34i)4-s + (−2.63 + 5.69i)5-s + (1.29 − 0.908i)6-s + (4.19 + 15.0i)7-s + (4.73 + 1.04i)8-s + (20.1 + 17.9i)9-s + (0.985 + 1.63i)10-s + (−43.2 − 40.9i)11-s + (0.611 + 41.0i)12-s + (−1.84 + 16.9i)13-s + (4.52 + 1.52i)14-s + (−23.3 + 22.7i)15-s + (−44.8 + 42.4i)16-s + (65.8 + 18.2i)17-s + ⋯
L(s)  = 1  + (0.0605 − 0.0892i)2-s + (0.934 + 0.356i)3-s + (0.365 + 0.918i)4-s + (−0.235 + 0.508i)5-s + (0.0883 − 0.0618i)6-s + (0.226 + 0.815i)7-s + (0.209 + 0.0460i)8-s + (0.746 + 0.665i)9-s + (0.0311 + 0.0518i)10-s + (−1.18 − 1.12i)11-s + (0.0147 + 0.988i)12-s + (−0.0394 + 0.362i)13-s + (0.0864 + 0.0291i)14-s + (−0.401 + 0.391i)15-s + (−0.700 + 0.663i)16-s + (0.939 + 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.138 - 0.990i$
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: \((2,\ 177,\ (\ :3/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56341 + 1.79772i\)
\(L(\frac12)\) \(\approx\) \(1.56341 + 1.79772i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.85 - 1.85i)T \)
59 \( 1 + (-152. + 426. i)T \)
good2 \( 1 + (-0.171 + 0.252i)T + (-2.96 - 7.43i)T^{2} \)
5 \( 1 + (2.63 - 5.69i)T + (-80.9 - 95.2i)T^{2} \)
7 \( 1 + (-4.19 - 15.0i)T + (-293. + 176. i)T^{2} \)
11 \( 1 + (43.2 + 40.9i)T + (72.0 + 1.32e3i)T^{2} \)
13 \( 1 + (1.84 - 16.9i)T + (-2.14e3 - 472. i)T^{2} \)
17 \( 1 + (-65.8 - 18.2i)T + (4.20e3 + 2.53e3i)T^{2} \)
19 \( 1 + (80.5 + 61.2i)T + (1.83e3 + 6.60e3i)T^{2} \)
23 \( 1 + (9.89 + 18.6i)T + (-6.82e3 + 1.00e4i)T^{2} \)
29 \( 1 + (-234. + 158. i)T + (9.02e3 - 2.26e4i)T^{2} \)
31 \( 1 + (135. + 178. i)T + (-7.96e3 + 2.87e4i)T^{2} \)
37 \( 1 + (-62.9 - 286. i)T + (-4.59e4 + 2.12e4i)T^{2} \)
41 \( 1 + (-383. - 203. i)T + (3.86e4 + 5.70e4i)T^{2} \)
43 \( 1 + (-377. - 398. i)T + (-4.30e3 + 7.93e4i)T^{2} \)
47 \( 1 + (290. - 134. i)T + (6.72e4 - 7.91e4i)T^{2} \)
53 \( 1 + (77.6 - 129. i)T + (-6.97e4 - 1.31e5i)T^{2} \)
61 \( 1 + (59.7 + 40.5i)T + (8.40e4 + 2.10e5i)T^{2} \)
67 \( 1 + (-204. + 927. i)T + (-2.72e5 - 1.26e5i)T^{2} \)
71 \( 1 + (204. + 442. i)T + (-2.31e5 + 2.72e5i)T^{2} \)
73 \( 1 + (-132. + 394. i)T + (-3.09e5 - 2.35e5i)T^{2} \)
79 \( 1 + (7.71 - 142. i)T + (-4.90e5 - 5.33e4i)T^{2} \)
83 \( 1 + (-19.6 + 119. i)T + (-5.41e5 - 1.82e5i)T^{2} \)
89 \( 1 + (417. + 615. i)T + (-2.60e5 + 6.54e5i)T^{2} \)
97 \( 1 + (-486. - 1.44e3i)T + (-7.26e5 + 5.52e5i)T^{2} \)
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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

−12.65531802613838473595197645307, −11.38674384316828216919884389986, −10.69958427078602003274609254837, −9.305276655030502911742716652922, −8.177057271950045251083222229807, −7.82045420457228662911751879746, −6.26509974313497936182893230330, −4.56970027271400843275988788269, −3.13834106635304570398808524469, −2.48899187785744037364216350145, 0.988632629997626934689409655173, 2.39655952685554569047111424131, 4.20680035163664071094438482873, 5.39675477099421848429402503593, 7.02827834370866012275897978290, 7.67607656515864840554945245722, 8.840783524319402677653563769825, 10.20749764011709744148548093572, 10.51314633638196008622278992992, 12.35101039250260144640574768849

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