Properties

Label 2-177-177.23-c1-0-5
Degree $2$
Conductor $177$
Sign $0.413 - 0.910i$
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  + (0.703 − 0.666i)2-s + (−0.733 + 1.56i)3-s + (−0.0574 + 1.05i)4-s + (−0.383 − 0.0629i)5-s + (0.529 + 1.59i)6-s + (−0.729 + 1.37i)7-s + (1.92 + 2.26i)8-s + (−1.92 − 2.30i)9-s + (−0.311 + 0.211i)10-s + (2.37 − 0.258i)11-s + (−1.62 − 0.867i)12-s + (−2.83 + 6.13i)13-s + (0.403 + 1.45i)14-s + (0.380 − 0.556i)15-s + (0.747 + 0.0812i)16-s + (5.86 − 3.10i)17-s + ⋯
L(s)  = 1  + (0.497 − 0.471i)2-s + (−0.423 + 0.905i)3-s + (−0.0287 + 0.529i)4-s + (−0.171 − 0.0281i)5-s + (0.216 + 0.650i)6-s + (−0.275 + 0.520i)7-s + (0.678 + 0.799i)8-s + (−0.641 − 0.767i)9-s + (−0.0986 + 0.0668i)10-s + (0.716 − 0.0779i)11-s + (−0.467 − 0.250i)12-s + (−0.787 + 1.70i)13-s + (0.107 + 0.388i)14-s + (0.0981 − 0.143i)15-s + (0.186 + 0.0203i)16-s + (1.42 − 0.753i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.413 - 0.910i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.413 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04017 + 0.670193i\)
\(L(\frac12)\) \(\approx\) \(1.04017 + 0.670193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.733 - 1.56i)T \)
59 \( 1 + (2.66 + 7.20i)T \)
good2 \( 1 + (-0.703 + 0.666i)T + (0.108 - 1.99i)T^{2} \)
5 \( 1 + (0.383 + 0.0629i)T + (4.73 + 1.59i)T^{2} \)
7 \( 1 + (0.729 - 1.37i)T + (-3.92 - 5.79i)T^{2} \)
11 \( 1 + (-2.37 + 0.258i)T + (10.7 - 2.36i)T^{2} \)
13 \( 1 + (2.83 - 6.13i)T + (-8.41 - 9.90i)T^{2} \)
17 \( 1 + (-5.86 + 3.10i)T + (9.54 - 14.0i)T^{2} \)
19 \( 1 + (-4.92 + 2.96i)T + (8.89 - 16.7i)T^{2} \)
23 \( 1 + (2.59 + 6.50i)T + (-16.6 + 15.8i)T^{2} \)
29 \( 1 + (1.89 - 1.99i)T + (-1.57 - 28.9i)T^{2} \)
31 \( 1 + (-4.19 + 6.96i)T + (-14.5 - 27.3i)T^{2} \)
37 \( 1 + (-3.34 - 2.84i)T + (5.98 + 36.5i)T^{2} \)
41 \( 1 + (-4.48 - 1.78i)T + (29.7 + 28.1i)T^{2} \)
43 \( 1 + (0.462 - 4.25i)T + (-41.9 - 9.24i)T^{2} \)
47 \( 1 + (-0.416 - 2.54i)T + (-44.5 + 15.0i)T^{2} \)
53 \( 1 + (5.01 + 3.39i)T + (19.6 + 49.2i)T^{2} \)
61 \( 1 + (2.11 + 2.23i)T + (-3.30 + 60.9i)T^{2} \)
67 \( 1 + (-2.81 + 2.39i)T + (10.8 - 66.1i)T^{2} \)
71 \( 1 + (-2.93 + 0.481i)T + (67.2 - 22.6i)T^{2} \)
73 \( 1 + (-1.02 + 0.285i)T + (62.5 - 37.6i)T^{2} \)
79 \( 1 + (9.43 + 2.07i)T + (71.6 + 33.1i)T^{2} \)
83 \( 1 + (-9.05 - 6.88i)T + (22.2 + 79.9i)T^{2} \)
89 \( 1 + (2.33 + 2.21i)T + (4.81 + 88.8i)T^{2} \)
97 \( 1 + (-4.04 - 1.12i)T + (83.1 + 50.0i)T^{2} \)
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

−12.45961268609884488639383215921, −11.75331469808754121866229818255, −11.41761687505166414321218084618, −9.769724325621720435050070642740, −9.219284121421340066776771963223, −7.77934564244921134179395101385, −6.35052171667583714464788589608, −4.91612498939595841912449543309, −4.05916263645484084463496400051, −2.75900961823990998722999510932, 1.18646447684680296980111284364, 3.58692831684469171167252696843, 5.40760892047661856455466234154, 5.91930979170807856136657129814, 7.34698654822139535871406854795, 7.79671124914182111903167394694, 9.794807342510475608858435413706, 10.45015160947898047959273458147, 11.84242069030800435035364396833, 12.58502685412764524129058110721

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