Properties

Label 2-552-23.18-c1-0-3
Degree $2$
Conductor $552$
Sign $0.820 - 0.571i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)3-s + (−0.321 + 2.23i)5-s + (0.387 + 0.847i)7-s + (−0.142 − 0.989i)9-s + (2.69 − 0.791i)11-s + (−0.948 + 2.07i)13-s + (1.47 + 1.70i)15-s + (0.0340 − 0.0218i)17-s + (7.12 + 4.58i)19-s + (0.894 + 0.262i)21-s + (−4.33 + 2.05i)23-s + (−0.0948 − 0.0278i)25-s + (−0.841 − 0.540i)27-s + (−1.27 + 0.822i)29-s + (4.09 + 4.72i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (−0.143 + 0.999i)5-s + (0.146 + 0.320i)7-s + (−0.0474 − 0.329i)9-s + (0.812 − 0.238i)11-s + (−0.263 + 0.576i)13-s + (0.381 + 0.440i)15-s + (0.00825 − 0.00530i)17-s + (1.63 + 1.05i)19-s + (0.195 + 0.0572i)21-s + (−0.903 + 0.428i)23-s + (−0.0189 − 0.00557i)25-s + (−0.161 − 0.104i)27-s + (−0.237 + 0.152i)29-s + (0.735 + 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59028 + 0.499715i\)
\(L(\frac12)\) \(\approx\) \(1.59028 + 0.499715i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (4.33 - 2.05i)T \)
good5 \( 1 + (0.321 - 2.23i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.387 - 0.847i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-2.69 + 0.791i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (0.948 - 2.07i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.0340 + 0.0218i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-7.12 - 4.58i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (1.27 - 0.822i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-4.09 - 4.72i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (0.406 + 2.82i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (0.537 - 3.73i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-7.20 + 8.31i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 + (3.20 + 7.02i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (1.68 - 3.69i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (8.43 + 9.73i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (2.80 + 0.822i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-2.60 - 0.765i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (4.28 + 2.75i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (1.17 - 2.58i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.773 - 5.38i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (2.23 - 2.57i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (0.0164 - 0.114i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.94573885345124013501741805152, −9.908239907449940520342182032883, −9.151973232459992552667484591611, −8.080757267454458663149523824204, −7.25704479196985363733231173764, −6.49628623814963617124385233982, −5.47475214620639329922210548627, −3.89521647058148058575331974574, −3.00825017493484071142703355542, −1.64832034829251260780909626289, 1.06114576218818220774911134006, 2.83053274506705573295095464839, 4.17993191006694174195845680032, 4.83537962737843837980708801085, 5.96758111398040706316063581512, 7.34402711603559343134313033173, 8.076382558587709306591799048952, 9.114160077545269869375951354460, 9.585706539081832812906571299870, 10.60635288137748250278050175376

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