Properties

Label 2-552-184.51-c1-0-35
Degree $2$
Conductor $552$
Sign $0.195 + 0.980i$
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  + (−0.620 + 1.27i)2-s + (0.654 − 0.755i)3-s + (−1.23 − 1.57i)4-s + (0.514 − 3.57i)5-s + (0.554 + 1.30i)6-s + (0.174 + 0.382i)7-s + (2.76 − 0.586i)8-s + (−0.142 − 0.989i)9-s + (4.23 + 2.87i)10-s + (0.516 + 1.75i)11-s + (−1.99 − 0.102i)12-s + (−5.06 − 2.31i)13-s + (−0.594 − 0.0152i)14-s + (−2.36 − 2.73i)15-s + (−0.970 + 3.88i)16-s + (0.555 + 0.863i)17-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)2-s + (0.378 − 0.436i)3-s + (−0.615 − 0.788i)4-s + (0.230 − 1.60i)5-s + (0.226 + 0.531i)6-s + (0.0660 + 0.144i)7-s + (0.978 − 0.207i)8-s + (−0.0474 − 0.329i)9-s + (1.33 + 0.908i)10-s + (0.155 + 0.530i)11-s + (−0.576 − 0.0295i)12-s + (−1.40 − 0.642i)13-s + (−0.158 − 0.00406i)14-s + (−0.611 − 0.705i)15-s + (−0.242 + 0.970i)16-s + (0.134 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812135 - 0.666552i\)
\(L(\frac12)\) \(\approx\) \(0.812135 - 0.666552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.620 - 1.27i)T \)
3 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (4.11 - 2.47i)T \)
good5 \( 1 + (-0.514 + 3.57i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.174 - 0.382i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-0.516 - 1.75i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (5.06 + 2.31i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.555 - 0.863i)T + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (-3.08 + 4.79i)T + (-7.89 - 17.2i)T^{2} \)
29 \( 1 + (3.86 + 6.01i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-5.29 + 4.58i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + (1.07 + 7.49i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.68 - 11.7i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.37 + 2.05i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + 3.17iT - 47T^{2} \)
53 \( 1 + (-5.08 - 11.1i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (0.527 - 1.15i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (-7.14 - 8.24i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-1.57 + 5.35i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-1.19 + 4.06i)T + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (1.18 + 0.760i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (0.967 - 2.11i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-15.9 + 2.28i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (4.62 + 4.01i)T + (12.6 + 88.0i)T^{2} \)
97 \( 1 + (8.56 + 1.23i)T + (93.0 + 27.3i)T^{2} \)
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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

−9.962633469317601557747810081874, −9.557644586604255391883854409358, −8.746527535981213072862335139147, −7.86798825934968700839080905988, −7.31532678481818139421661756389, −5.93727211396500448527373828743, −5.14207308687950508542747208414, −4.30651999292521261518450671309, −2.12639330277774153873872088178, −0.66665671206797121218430684228, 2.05308300792522992283114304157, 3.04364265257926551951047620929, 3.82974455262924023011760550661, 5.18792581382723010504258607137, 6.72432044785103574901611209124, 7.50571609493725655418715942061, 8.471658505793116483523604152131, 9.636384763721206519052216122453, 10.11560642719626468719013932677, 10.74530314018794471483108161478

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