Properties

Label 2-177-177.38-c1-0-10
Degree $2$
Conductor $177$
Sign $0.554 - 0.831i$
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  + (1.60 + 0.964i)2-s + (1.54 + 0.774i)3-s + (0.701 + 1.32i)4-s + (−0.0291 + 0.267i)5-s + (1.73 + 2.73i)6-s + (−3.96 − 1.33i)7-s + (0.0509 − 0.939i)8-s + (1.79 + 2.40i)9-s + (−0.304 + 0.401i)10-s + (−1.25 − 1.84i)11-s + (0.0617 + 2.59i)12-s + (0.0885 + 0.0934i)13-s + (−5.05 − 5.95i)14-s + (−0.252 + 0.392i)15-s + (2.66 − 3.93i)16-s + (−0.877 − 2.60i)17-s + ⋯
L(s)  = 1  + (1.13 + 0.681i)2-s + (0.894 + 0.447i)3-s + (0.350 + 0.661i)4-s + (−0.0130 + 0.119i)5-s + (0.708 + 1.11i)6-s + (−1.49 − 0.504i)7-s + (0.0180 − 0.332i)8-s + (0.599 + 0.800i)9-s + (−0.0964 + 0.126i)10-s + (−0.377 − 0.557i)11-s + (0.0178 + 0.748i)12-s + (0.0245 + 0.0259i)13-s + (−1.35 − 1.59i)14-s + (−0.0652 + 0.101i)15-s + (0.666 − 0.983i)16-s + (−0.212 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.554 - 0.831i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.554 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95675 + 1.04680i\)
\(L(\frac12)\) \(\approx\) \(1.95675 + 1.04680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.54 - 0.774i)T \)
59 \( 1 + (1.76 + 7.47i)T \)
good2 \( 1 + (-1.60 - 0.964i)T + (0.936 + 1.76i)T^{2} \)
5 \( 1 + (0.0291 - 0.267i)T + (-4.88 - 1.07i)T^{2} \)
7 \( 1 + (3.96 + 1.33i)T + (5.57 + 4.23i)T^{2} \)
11 \( 1 + (1.25 + 1.84i)T + (-4.07 + 10.2i)T^{2} \)
13 \( 1 + (-0.0885 - 0.0934i)T + (-0.703 + 12.9i)T^{2} \)
17 \( 1 + (0.877 + 2.60i)T + (-13.5 + 10.2i)T^{2} \)
19 \( 1 + (1.29 - 7.88i)T + (-18.0 - 6.06i)T^{2} \)
23 \( 1 + (-0.724 + 2.61i)T + (-19.7 - 11.8i)T^{2} \)
29 \( 1 + (-3.17 - 5.27i)T + (-13.5 + 25.6i)T^{2} \)
31 \( 1 + (5.97 - 0.979i)T + (29.3 - 9.89i)T^{2} \)
37 \( 1 + (6.76 - 0.366i)T + (36.7 - 4.00i)T^{2} \)
41 \( 1 + (-6.10 + 1.69i)T + (35.1 - 21.1i)T^{2} \)
43 \( 1 + (2.60 + 1.76i)T + (15.9 + 39.9i)T^{2} \)
47 \( 1 + (-5.39 + 0.586i)T + (45.9 - 10.1i)T^{2} \)
53 \( 1 + (5.75 + 7.57i)T + (-14.1 + 51.0i)T^{2} \)
61 \( 1 + (0.611 - 1.01i)T + (-28.5 - 53.8i)T^{2} \)
67 \( 1 + (-6.57 - 0.356i)T + (66.6 + 7.24i)T^{2} \)
71 \( 1 + (-0.857 - 7.88i)T + (-69.3 + 15.2i)T^{2} \)
73 \( 1 + (3.64 - 3.09i)T + (11.8 - 72.0i)T^{2} \)
79 \( 1 + (1.78 + 4.48i)T + (-57.3 + 54.3i)T^{2} \)
83 \( 1 + (3.74 + 1.73i)T + (53.7 + 63.2i)T^{2} \)
89 \( 1 + (8.05 - 4.84i)T + (41.6 - 78.6i)T^{2} \)
97 \( 1 + (-7.78 - 6.61i)T + (15.6 + 95.7i)T^{2} \)
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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

−12.97496688092766629782104534331, −12.59465239187711034099864518030, −10.61377689808500799728613613477, −9.896292605054466693575416021542, −8.757327407977358167292326841730, −7.36788522036872896930467020678, −6.50423079946753138961059808835, −5.23461879751395438167923063945, −3.83716183312774455437745919312, −3.13396459932246597348674885117, 2.39441920027455136269459854247, 3.23507274410808231450835716068, 4.51763858133351155346128127166, 6.05958252581371253648780327649, 7.17144911199078304895671993816, 8.656980325886888818977768918285, 9.484068638718543014446839857762, 10.73508641334642589315473730412, 12.07559138968474597369962133417, 12.88719707340709465069526762513

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