Properties

Degree $2$
Conductor $1152$
Sign $-0.946 + 0.324i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.32 + 3.32i)5-s + 4.04·7-s + (−6.82 + 6.82i)11-s + (−4.29 + 4.29i)13-s − 30.1·17-s + (−19.7 − 19.7i)19-s − 28.2·23-s − 2.86i·25-s + (−21.3 + 21.3i)29-s − 38.0i·31-s + (13.4 + 13.4i)35-s + (42.8 + 42.8i)37-s − 48.2i·41-s + (32.6 − 32.6i)43-s − 15.8i·47-s + ⋯
L(s)  = 1  + (0.665 + 0.665i)5-s + 0.577·7-s + (−0.620 + 0.620i)11-s + (−0.330 + 0.330i)13-s − 1.77·17-s + (−1.03 − 1.03i)19-s − 1.22·23-s − 0.114i·25-s + (−0.736 + 0.736i)29-s − 1.22i·31-s + (0.384 + 0.384i)35-s + (1.15 + 1.15i)37-s − 1.17i·41-s + (0.759 − 0.759i)43-s − 0.336i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.324i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.946 + 0.324i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.946 + 0.324i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.08878654321\)
\(L(\frac12)\) \(\approx\) \(0.08878654321\)
\(L(2)\) 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 \( 1 + (-3.32 - 3.32i)T + 25iT^{2} \)
7 \( 1 - 4.04T + 49T^{2} \)
11 \( 1 + (6.82 - 6.82i)T - 121iT^{2} \)
13 \( 1 + (4.29 - 4.29i)T - 169iT^{2} \)
17 \( 1 + 30.1T + 289T^{2} \)
19 \( 1 + (19.7 + 19.7i)T + 361iT^{2} \)
23 \( 1 + 28.2T + 529T^{2} \)
29 \( 1 + (21.3 - 21.3i)T - 841iT^{2} \)
31 \( 1 + 38.0iT - 961T^{2} \)
37 \( 1 + (-42.8 - 42.8i)T + 1.36e3iT^{2} \)
41 \( 1 + 48.2iT - 1.68e3T^{2} \)
43 \( 1 + (-32.6 + 32.6i)T - 1.84e3iT^{2} \)
47 \( 1 + 15.8iT - 2.20e3T^{2} \)
53 \( 1 + (0.476 + 0.476i)T + 2.80e3iT^{2} \)
59 \( 1 + (9.97 - 9.97i)T - 3.48e3iT^{2} \)
61 \( 1 + (37.9 - 37.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (-20.0 - 20.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 40.0T + 5.04e3T^{2} \)
73 \( 1 - 30.8iT - 5.32e3T^{2} \)
79 \( 1 + 130. iT - 6.24e3T^{2} \)
83 \( 1 + (-2.26 - 2.26i)T + 6.88e3iT^{2} \)
89 \( 1 - 72.2iT - 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.10902098431295131694792878845, −9.284329776633406990452927201430, −8.475553218497440256016039974455, −7.48482295858626990659930205563, −6.71715316644530665893846607362, −5.98235492919622982078019098077, −4.81194756448195241208156884128, −4.14944569852426114154644632257, −2.40429882933348694771599425604, −2.11725307871223062337197424017, 0.02338462871902627541142655691, 1.62740779865280792275908665748, 2.50705895694756639580366076213, 4.03424618468746361932059193130, 4.84495939206462632841996395446, 5.77843943687443129289970701179, 6.40602148378441493797768026674, 7.78760886313413089963619323699, 8.281756574757706184001671760584, 9.150443827894897022043174917836

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