↑

Properties

Label 2-972-27.16-c1-0-8
Degree $2$
Conductor $972$
Sign $0.179 + 0.983i$
Analytic cond. $7.76145$
Root an. cond. $2.78593$
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  + (−0.558 − 0.203i)5-s + (0.495 − 2.81i)7-s + (0.383 − 0.139i)11-s + (0.148 − 0.124i)13-s + (3.66 + 6.34i)17-s + (2.06 − 3.57i)19-s + (−0.707 − 4.01i)23-s + (−3.56 − 2.98i)25-s + (−7.88 − 6.61i)29-s + (−1.08 − 6.14i)31-s + (−0.848 + 1.46i)35-s + (−2.88 − 4.99i)37-s + (6.46 − 5.42i)41-s + (5.62 − 2.04i)43-s + (−1.95 + 11.0i)47-s + ⋯
L(s)  = 1  + (−0.249 − 0.0908i)5-s + (0.187 − 1.06i)7-s + (0.115 − 0.0420i)11-s + (0.0412 − 0.0346i)13-s + (0.888 + 1.53i)17-s + (0.473 − 0.820i)19-s + (−0.147 − 0.836i)23-s + (−0.712 − 0.597i)25-s + (−1.46 − 1.22i)29-s + (−0.194 − 1.10i)31-s + (−0.143 + 0.248i)35-s + (−0.474 − 0.821i)37-s + (1.00 − 0.846i)41-s + (0.858 − 0.312i)43-s + (−0.285 + 1.61i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(972\)    =    \(2^{2} \cdot 3^{5}\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(7.76145\)
Root analytic conductor: \(2.78593\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{972} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 972,\ (\ :1/2),\ 0.179 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05929 - 0.883478i\)
\(L(\frac12)\) \(\approx\) \(1.05929 - 0.883478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.558 + 0.203i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-0.495 + 2.81i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.383 + 0.139i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.148 + 0.124i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-3.66 - 6.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.06 + 3.57i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.707 + 4.01i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (7.88 + 6.61i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (1.08 + 6.14i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.88 + 4.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.46 + 5.42i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.62 + 2.04i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.95 - 11.0i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.42T + 53T^{2} \)
59 \( 1 + (-4.40 - 1.60i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.758 + 4.30i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-9.31 + 7.81i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (4.09 + 7.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.96 + 3.41i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.91 - 5.79i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.45 + 5.41i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (2.38 - 4.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.27 - 2.64i)T + (74.3 - 62.3i)T^{2} \)
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

−9.876884957054553464335506297662, −9.051741495938333725547532916893, −7.82655784965445675104090923731, −7.63335366408849037062084094285, −6.37163822406692979299604593795, −5.57333856825554454577174684138, −4.19382754376064492293447097558, −3.80567043865691667053632985066, −2.18921432263825041609812875828, −0.67518766667743382361820670907, 1.52381808949912930912537669900, 2.88774097066748535357947002004, 3.78568305550360402084382158891, 5.37263185576711569918688569292, 5.49282752886924398812752086115, 6.98414975473219050121646381016, 7.62452448251176485196494929289, 8.591423502389061102190565855419, 9.393887867544071683554397166586, 9.996719222919386382233286297189

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