Properties

Label 2-177-177.32-c1-0-10
Degree $2$
Conductor $177$
Sign $0.995 + 0.0935i$
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  + (0.110 − 0.397i)2-s + (1.71 + 0.225i)3-s + (1.56 + 0.943i)4-s + (−2.16 − 0.117i)5-s + (0.279 − 0.658i)6-s + (0.253 − 1.54i)7-s + (1.14 − 1.08i)8-s + (2.89 + 0.773i)9-s + (−0.286 + 0.849i)10-s + (−0.507 + 0.957i)11-s + (2.47 + 1.97i)12-s + (−2.53 + 1.00i)13-s + (−0.587 − 0.271i)14-s + (−3.69 − 0.689i)15-s + (1.40 + 2.65i)16-s + (−1.49 + 0.245i)17-s + ⋯
L(s)  = 1  + (0.0781 − 0.281i)2-s + (0.991 + 0.129i)3-s + (0.783 + 0.471i)4-s + (−0.969 − 0.0525i)5-s + (0.114 − 0.268i)6-s + (0.0958 − 0.584i)7-s + (0.405 − 0.384i)8-s + (0.966 + 0.257i)9-s + (−0.0904 + 0.268i)10-s + (−0.153 + 0.288i)11-s + (0.715 + 0.569i)12-s + (−0.702 + 0.279i)13-s + (−0.157 − 0.0726i)14-s + (−0.953 − 0.178i)15-s + (0.351 + 0.663i)16-s + (−0.363 + 0.0595i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.995 + 0.0935i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.995 + 0.0935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62765 - 0.0763037i\)
\(L(\frac12)\) \(\approx\) \(1.62765 - 0.0763037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 - 0.225i)T \)
59 \( 1 + (3.19 - 6.98i)T \)
good2 \( 1 + (-0.110 + 0.397i)T + (-1.71 - 1.03i)T^{2} \)
5 \( 1 + (2.16 + 0.117i)T + (4.97 + 0.540i)T^{2} \)
7 \( 1 + (-0.253 + 1.54i)T + (-6.63 - 2.23i)T^{2} \)
11 \( 1 + (0.507 - 0.957i)T + (-6.17 - 9.10i)T^{2} \)
13 \( 1 + (2.53 - 1.00i)T + (9.43 - 8.94i)T^{2} \)
17 \( 1 + (1.49 - 0.245i)T + (16.1 - 5.42i)T^{2} \)
19 \( 1 + (2.26 + 2.66i)T + (-3.07 + 18.7i)T^{2} \)
23 \( 1 + (-1.57 + 1.19i)T + (6.15 - 22.1i)T^{2} \)
29 \( 1 + (3.75 - 1.04i)T + (24.8 - 14.9i)T^{2} \)
31 \( 1 + (4.33 + 3.68i)T + (5.01 + 30.5i)T^{2} \)
37 \( 1 + (-5.65 + 5.96i)T + (-2.00 - 36.9i)T^{2} \)
41 \( 1 + (6.36 - 8.37i)T + (-10.9 - 39.5i)T^{2} \)
43 \( 1 + (8.66 - 4.59i)T + (24.1 - 35.5i)T^{2} \)
47 \( 1 + (0.568 + 10.4i)T + (-46.7 + 5.08i)T^{2} \)
53 \( 1 + (-0.757 - 2.24i)T + (-42.1 + 32.0i)T^{2} \)
61 \( 1 + (-5.58 - 1.55i)T + (52.2 + 31.4i)T^{2} \)
67 \( 1 + (-5.58 - 5.89i)T + (-3.62 + 66.9i)T^{2} \)
71 \( 1 + (-12.8 + 0.697i)T + (70.5 - 7.67i)T^{2} \)
73 \( 1 + (-0.608 + 1.31i)T + (-47.2 - 55.6i)T^{2} \)
79 \( 1 + (-2.11 + 3.11i)T + (-29.2 - 73.3i)T^{2} \)
83 \( 1 + (-10.1 - 2.23i)T + (75.3 + 34.8i)T^{2} \)
89 \( 1 + (2.28 + 8.21i)T + (-76.2 + 45.8i)T^{2} \)
97 \( 1 + (0.449 + 0.971i)T + (-62.7 + 73.9i)T^{2} \)
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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

−12.74258408470031573163363808486, −11.66084467248064568347854775265, −10.83196100625567511774405415710, −9.721254654114565035837659951691, −8.413827288174108957686076155436, −7.53269151776205490597310461261, −6.90716746841321295237688900469, −4.50235138243206881971671474828, −3.60289227185144441936111610725, −2.23366195071881073628060205869, 2.14433916952994646589960645709, 3.53358534280789562132478900341, 5.15048145054341172386597473135, 6.64986329886055734910254095268, 7.63440137206181325676690246857, 8.365301256827596909919280986278, 9.629391779088033259513111280066, 10.76936981781364657470957174760, 11.78936292259535069702228992131, 12.65246847983175878988224498781

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