↑

Properties

Label 2-1176-21.17-c1-0-34
Degree $2$
Conductor $1176$
Sign $-0.526 + 0.850i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (0.651 − 1.60i)3-s + (0.661 + 1.14i)5-s + (−2.15 − 2.09i)9-s + (−2.43 − 1.40i)11-s − 2.52i·13-s + (2.27 − 0.315i)15-s + (1.63 − 2.82i)17-s + (−4.96 + 2.86i)19-s + (3.68 − 2.12i)23-s + (1.62 − 2.81i)25-s + (−4.75 + 2.09i)27-s − 4.75i·29-s + (−7.88 − 4.55i)31-s + (−3.84 + 2.99i)33-s + (−5.63 − 9.75i)37-s + ⋯
L(s)  = 1  + (0.376 − 0.926i)3-s + (0.296 + 0.512i)5-s + (−0.717 − 0.696i)9-s + (−0.733 − 0.423i)11-s − 0.701i·13-s + (0.586 − 0.0814i)15-s + (0.395 − 0.685i)17-s + (−1.13 + 0.658i)19-s + (0.769 − 0.443i)23-s + (0.324 − 0.562i)25-s + (−0.915 + 0.402i)27-s − 0.883i·29-s + (−1.41 − 0.817i)31-s + (−0.668 + 0.520i)33-s + (−0.925 − 1.60i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.526 + 0.850i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.448421748\)
\(L(\frac12)\) \(\approx\) \(1.448421748\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.651 + 1.60i)T \)
7 \( 1 \)
good5 \( 1 + (-0.661 - 1.14i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.43 + 1.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.52iT - 13T^{2} \)
17 \( 1 + (-1.63 + 2.82i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.68 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.75iT - 29T^{2} \)
31 \( 1 + (7.88 + 4.55i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.63 + 9.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + (0.744 + 1.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.4 - 6.02i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.44 - 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.03 - 1.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0654 + 0.113i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.875iT - 71T^{2} \)
73 \( 1 + (5.58 + 3.22i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.96 + 5.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.398T + 83T^{2} \)
89 \( 1 + (2.87 + 4.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.36iT - 97T^{2} \)
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

−9.311819101689533463415315341954, −8.635240197260128946536650463082, −7.64763680768917433983711718143, −7.26342133377657580612526056254, −6.02017072759201836726033289973, −5.66224453536968706153228342290, −4.09020370855509972053802202052, −2.86957828602365403888442929471, −2.25020149487860969712029020725, −0.57174012102674376780581102164, 1.74566580136622264027197686292, 2.94251697141772623252287526855, 4.02400088095803586661444355245, 4.92905246319146041481101129778, 5.50296032057059588625847467274, 6.76367576866281042040491337166, 7.71793619128463225856179640270, 8.808540919070097981157943872297, 9.043083305429827773265677765373, 10.01471656917725320021207761256

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