Properties

Label 2-552-184.51-c1-0-1
Degree $2$
Conductor $552$
Sign $0.317 - 0.948i$
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  + (−0.908 − 1.08i)2-s + (−0.654 + 0.755i)3-s + (−0.349 + 1.96i)4-s + (0.242 − 1.68i)5-s + (1.41 + 0.0233i)6-s + (2.11 + 4.62i)7-s + (2.45 − 1.40i)8-s + (−0.142 − 0.989i)9-s + (−2.05 + 1.27i)10-s + (0.346 + 1.17i)11-s + (−1.25 − 1.55i)12-s + (−4.65 − 2.12i)13-s + (3.09 − 6.49i)14-s + (1.11 + 1.29i)15-s + (−3.75 − 1.37i)16-s + (0.812 + 1.26i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.378 + 0.436i)3-s + (−0.174 + 0.984i)4-s + (0.108 − 0.755i)5-s + (0.577 + 0.00954i)6-s + (0.799 + 1.74i)7-s + (0.867 − 0.498i)8-s + (−0.0474 − 0.329i)9-s + (−0.649 + 0.402i)10-s + (0.104 + 0.355i)11-s + (−0.363 − 0.448i)12-s + (−1.29 − 0.589i)13-s + (0.827 − 1.73i)14-s + (0.288 + 0.333i)15-s + (−0.938 − 0.344i)16-s + (0.197 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.317 - 0.948i$
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: \((2,\ 552,\ (\ :1/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588788 + 0.423597i\)
\(L(\frac12)\) \(\approx\) \(0.588788 + 0.423597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.908 + 1.08i)T \)
3 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (4.59 + 1.37i)T \)
good5 \( 1 + (-0.242 + 1.68i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-2.11 - 4.62i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-0.346 - 1.17i)T + (-9.25 + 5.94i)T^{2} \)
13 \( 1 + (4.65 + 2.12i)T + (8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.812 - 1.26i)T + (-7.06 + 15.4i)T^{2} \)
19 \( 1 + (3.59 - 5.59i)T + (-7.89 - 17.2i)T^{2} \)
29 \( 1 + (-1.37 - 2.13i)T + (-12.0 + 26.3i)T^{2} \)
31 \( 1 + (-3.02 + 2.62i)T + (4.41 - 30.6i)T^{2} \)
37 \( 1 + (-0.401 - 2.79i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.80 - 12.5i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-3.95 - 3.42i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 - 8.25iT - 47T^{2} \)
53 \( 1 + (4.54 + 9.94i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (0.636 - 1.39i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (1.52 + 1.75i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-0.0236 + 0.0806i)T + (-56.3 - 36.2i)T^{2} \)
71 \( 1 + (-2.59 + 8.84i)T + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (-12.7 - 8.22i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (2.81 - 6.16i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.205 + 0.0294i)T + (79.6 - 23.3i)T^{2} \)
89 \( 1 + (1.02 + 0.885i)T + (12.6 + 88.0i)T^{2} \)
97 \( 1 + (10.6 + 1.52i)T + (93.0 + 27.3i)T^{2} \)
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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

−10.97734248326716817563799188684, −9.884255553413191179286578560979, −9.452495877706180537177933771407, −8.267177240480998729046605320408, −8.072294372692340130009346105695, −6.23734535688839843201071065856, −5.13257782030000169424692441516, −4.44929887756917895783003085310, −2.78143348360763978071937555439, −1.68080147911828033289005112577, 0.54320920467366113513173381864, 2.14937549255636923252734020235, 4.22476286389556073374078872535, 5.09971689571448801589573620963, 6.47551549366767975495536509359, 7.14052607219966748439821786638, 7.54465770762494140366500396580, 8.660353495059623182821748604855, 9.871767379992118505254075900831, 10.62044361030591665130519038227

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