Properties

Label 2-177-59.10-c2-0-7
Degree $2$
Conductor $177$
Sign $-0.393 - 0.919i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.27 + 1.30i)2-s + (−1.72 − 0.187i)3-s + (6.09 + 5.77i)4-s + (−5.84 + 6.87i)5-s + (−5.38 − 2.85i)6-s + (−0.280 − 0.168i)7-s + (6.50 + 14.0i)8-s + (2.92 + 0.644i)9-s + (−28.0 + 14.8i)10-s + (5.71 + 0.310i)11-s + (−9.42 − 11.0i)12-s + (0.0412 + 0.187i)13-s + (−0.696 − 0.916i)14-s + (11.3 − 10.7i)15-s + (1.13 + 20.9i)16-s + (23.2 − 13.9i)17-s + ⋯
L(s)  = 1  + (1.63 + 0.651i)2-s + (−0.573 − 0.0624i)3-s + (1.52 + 1.44i)4-s + (−1.16 + 1.37i)5-s + (−0.898 − 0.476i)6-s + (−0.0400 − 0.0240i)7-s + (0.813 + 1.75i)8-s + (0.325 + 0.0716i)9-s + (−2.80 + 1.48i)10-s + (0.519 + 0.0281i)11-s + (−0.785 − 0.924i)12-s + (0.00317 + 0.0144i)13-s + (−0.0497 − 0.0654i)14-s + (0.756 − 0.716i)15-s + (0.0709 + 1.30i)16-s + (1.36 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ -0.393 - 0.919i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42400 + 2.15954i\)
\(L(\frac12)\) \(\approx\) \(1.42400 + 2.15954i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.187i)T \)
59 \( 1 + (48.5 - 33.4i)T \)
good2 \( 1 + (-3.27 - 1.30i)T + (2.90 + 2.75i)T^{2} \)
5 \( 1 + (5.84 - 6.87i)T + (-4.04 - 24.6i)T^{2} \)
7 \( 1 + (0.280 + 0.168i)T + (22.9 + 43.2i)T^{2} \)
11 \( 1 + (-5.71 - 0.310i)T + (120. + 13.0i)T^{2} \)
13 \( 1 + (-0.0412 - 0.187i)T + (-153. + 70.9i)T^{2} \)
17 \( 1 + (-23.2 + 13.9i)T + (135. - 255. i)T^{2} \)
19 \( 1 + (7.91 - 28.5i)T + (-309. - 186. i)T^{2} \)
23 \( 1 + (-24.1 + 16.3i)T + (195. - 491. i)T^{2} \)
29 \( 1 + (-14.1 - 35.5i)T + (-610. + 578. i)T^{2} \)
31 \( 1 + (-0.724 + 0.201i)T + (823. - 495. i)T^{2} \)
37 \( 1 + (-13.8 + 29.8i)T + (-886. - 1.04e3i)T^{2} \)
41 \( 1 + (3.61 - 5.32i)T + (-622. - 1.56e3i)T^{2} \)
43 \( 1 + (64.4 - 3.49i)T + (1.83e3 - 199. i)T^{2} \)
47 \( 1 + (-16.2 + 13.7i)T + (357. - 2.17e3i)T^{2} \)
53 \( 1 + (-24.2 + 45.7i)T + (-1.57e3 - 2.32e3i)T^{2} \)
61 \( 1 + (64.0 + 25.5i)T + (2.70e3 + 2.55e3i)T^{2} \)
67 \( 1 + (-21.6 - 46.8i)T + (-2.90e3 + 3.42e3i)T^{2} \)
71 \( 1 + (-14.8 - 17.5i)T + (-815. + 4.97e3i)T^{2} \)
73 \( 1 + (-82.4 - 108. i)T + (-1.42e3 + 5.13e3i)T^{2} \)
79 \( 1 + (-58.9 + 6.41i)T + (6.09e3 - 1.34e3i)T^{2} \)
83 \( 1 + (1.43 + 4.25i)T + (-5.48e3 + 4.16e3i)T^{2} \)
89 \( 1 + (-88.8 + 35.4i)T + (5.75e3 - 5.44e3i)T^{2} \)
97 \( 1 + (37.5 - 49.3i)T + (-2.51e3 - 9.06e3i)T^{2} \)
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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

−12.61489155569395708747120786577, −12.00613072670133119324279576070, −11.28063560296511490964260926454, −10.26148203805571629472151672851, −8.015156169877987867179742379632, −7.07902172906928239115334147491, −6.48938731228491646152830257988, −5.22961201160374856109517554936, −3.92005552732343133003679693923, −3.10182080512900063587983182057, 1.13067931406141057301711514210, 3.42308907315879941470972680548, 4.48692842018319258104094308724, 5.13735129738309650806845401306, 6.38262873434443255547915661860, 7.87057169152814583042361560153, 9.280786542951015603954428758626, 10.77593790960358165251871555159, 11.67080065331053814812067993892, 12.14980374137509347364601904330

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