↑

Properties

Label 2-177-177.176-c1-0-6
Degree $2$
Conductor $177$
Sign $0.481 - 0.876i$
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  + 1.93·2-s + (−1.34 + 1.09i)3-s + 1.74·4-s + 3.21i·5-s + (−2.59 + 2.12i)6-s + 0.254·7-s − 0.491·8-s + (0.594 − 2.94i)9-s + 6.23i·10-s + 5.68·11-s + (−2.34 + 1.91i)12-s − 2.66i·13-s + 0.491·14-s + (−3.53 − 4.31i)15-s − 4.44·16-s − 4.03i·17-s + ⋯
L(s)  = 1  + 1.36·2-s + (−0.774 + 0.633i)3-s + 0.872·4-s + 1.43i·5-s + (−1.05 + 0.866i)6-s + 0.0960·7-s − 0.173·8-s + (0.198 − 0.980i)9-s + 1.97i·10-s + 1.71·11-s + (−0.675 + 0.552i)12-s − 0.738i·13-s + 0.131·14-s + (−0.911 − 1.11i)15-s − 1.11·16-s − 0.979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(1.41335\)
Root analytic conductor: \(1.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53463 + 0.907944i\)
\(L(\frac12)\) \(\approx\) \(1.53463 + 0.907944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.34 - 1.09i)T \)
59 \( 1 + (-7.12 + 2.86i)T \)
good2 \( 1 - 1.93T + 2T^{2} \)
5 \( 1 - 3.21iT - 5T^{2} \)
7 \( 1 - 0.254T + 7T^{2} \)
11 \( 1 - 5.68T + 11T^{2} \)
13 \( 1 + 2.66iT - 13T^{2} \)
17 \( 1 + 4.03iT - 17T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 - 3.25T + 23T^{2} \)
29 \( 1 + 3.56iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + 5.15iT - 37T^{2} \)
41 \( 1 - 7.25iT - 41T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + 1.16iT - 53T^{2} \)
61 \( 1 - 0.676iT - 61T^{2} \)
67 \( 1 - 2.66iT - 67T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 + 7.95T + 89T^{2} \)
97 \( 1 + 4.47iT - 97T^{2} \)
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

−12.80924098244947319233606149011, −11.73566683614342473950849654273, −11.28597159003543621968849415373, −10.22821682279590894436525486929, −9.083335351368124513220903426749, −6.91603822781302418863104118942, −6.41076694317030642217371086399, −5.23344888611309216641120488990, −3.99987280361672570519568626016, −3.07222543395899032685408247983, 1.55713254441623194389028352354, 4.07201939147727172577406512633, 4.80916649247499712683415799112, 5.97089529695821154609300320714, 6.73918470219920389501117260155, 8.431184151933579116809129088233, 9.388266434690057954717921151559, 11.25127103025428518576851875423, 11.89679111249679717621057164924, 12.66365080988688432395073896907

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