Properties

Label 2-177-59.4-c1-0-2
Degree $2$
Conductor $177$
Sign $0.921 - 0.389i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.38 − 0.150i)2-s + (0.647 − 0.762i)3-s + (−0.0581 − 0.0128i)4-s + (1.49 + 1.13i)5-s + (−1.01 + 0.957i)6-s + (−1.93 + 4.85i)7-s + (2.71 + 0.916i)8-s + (−0.161 − 0.986i)9-s + (−1.90 − 1.80i)10-s + (4.37 − 2.02i)11-s + (−0.0474 + 0.0360i)12-s + (0.300 − 1.83i)13-s + (3.40 − 6.42i)14-s + (1.83 − 0.404i)15-s + (−3.51 − 1.62i)16-s + (0.912 + 2.28i)17-s + ⋯
L(s)  = 1  + (−0.979 − 0.106i)2-s + (0.373 − 0.440i)3-s + (−0.0290 − 0.00640i)4-s + (0.669 + 0.508i)5-s + (−0.412 + 0.391i)6-s + (−0.730 + 1.83i)7-s + (0.961 + 0.323i)8-s + (−0.0539 − 0.328i)9-s + (−0.601 − 0.569i)10-s + (1.31 − 0.610i)11-s + (−0.0136 + 0.0104i)12-s + (0.0833 − 0.508i)13-s + (0.911 − 1.71i)14-s + (0.473 − 0.104i)15-s + (−0.879 − 0.407i)16-s + (0.221 + 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.921 - 0.389i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.921 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794269 + 0.160994i\)
\(L(\frac12)\) \(\approx\) \(0.794269 + 0.160994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.647 + 0.762i)T \)
59 \( 1 + (-1.04 + 7.60i)T \)
good2 \( 1 + (1.38 + 0.150i)T + (1.95 + 0.429i)T^{2} \)
5 \( 1 + (-1.49 - 1.13i)T + (1.33 + 4.81i)T^{2} \)
7 \( 1 + (1.93 - 4.85i)T + (-5.08 - 4.81i)T^{2} \)
11 \( 1 + (-4.37 + 2.02i)T + (7.12 - 8.38i)T^{2} \)
13 \( 1 + (-0.300 + 1.83i)T + (-12.3 - 4.15i)T^{2} \)
17 \( 1 + (-0.912 - 2.28i)T + (-12.3 + 11.6i)T^{2} \)
19 \( 1 + (-3.76 - 5.55i)T + (-7.03 + 17.6i)T^{2} \)
23 \( 1 + (-0.0327 + 0.603i)T + (-22.8 - 2.48i)T^{2} \)
29 \( 1 + (-0.0486 + 0.00529i)T + (28.3 - 6.23i)T^{2} \)
31 \( 1 + (0.180 - 0.265i)T + (-11.4 - 28.7i)T^{2} \)
37 \( 1 + (7.00 - 2.36i)T + (29.4 - 22.3i)T^{2} \)
41 \( 1 + (0.238 + 4.40i)T + (-40.7 + 4.43i)T^{2} \)
43 \( 1 + (-4.90 - 2.26i)T + (27.8 + 32.7i)T^{2} \)
47 \( 1 + (-2.52 + 1.92i)T + (12.5 - 45.2i)T^{2} \)
53 \( 1 + (7.42 - 7.03i)T + (2.86 - 52.9i)T^{2} \)
61 \( 1 + (6.24 + 0.679i)T + (59.5 + 13.1i)T^{2} \)
67 \( 1 + (0.774 + 0.261i)T + (53.3 + 40.5i)T^{2} \)
71 \( 1 + (-3.23 + 2.45i)T + (18.9 - 68.4i)T^{2} \)
73 \( 1 + (-5.38 + 10.1i)T + (-40.9 - 60.4i)T^{2} \)
79 \( 1 + (6.39 + 7.53i)T + (-12.7 + 77.9i)T^{2} \)
83 \( 1 + (0.156 - 0.0942i)T + (38.8 - 73.3i)T^{2} \)
89 \( 1 + (-6.23 + 0.678i)T + (86.9 - 19.1i)T^{2} \)
97 \( 1 + (-7.73 - 14.5i)T + (-54.4 + 80.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.58773762272593893390321162235, −11.84151411035490301142486634465, −10.42808811009165969350427612159, −9.472796411305064240709118393093, −8.912838907627138126471210389173, −7.998682362854247222152545627081, −6.41743752042588213563457367832, −5.67021822383178063259135354372, −3.28936136838158232760414213912, −1.80195320109106594620790645828, 1.18060532943869844531223850107, 3.76328087821203639657930360792, 4.74235390910476106574641071691, 6.81750039576683964434647743077, 7.45923587118574890382099554654, 9.053985742736817662172383892902, 9.486522966139363255713849214081, 10.14833075225460421397666533355, 11.27970010190333980434215549371, 12.90080228793749366125512640892

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