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Properties

Label 2-550-11.4-c1-0-11
Degree $2$
Conductor $550$
Sign $0.530 + 0.847i$
Analytic cond. $4.39177$
Root an. cond. $2.09565$
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  + (−0.809 − 0.587i)2-s + (0.381 − 1.17i)3-s + (0.309 + 0.951i)4-s + (−1 + 0.726i)6-s + (−0.190 − 0.587i)7-s + (0.309 − 0.951i)8-s + (1.19 + 0.865i)9-s + (1.69 + 2.85i)11-s + 1.23·12-s + (1.92 + 1.40i)13-s + (−0.190 + 0.587i)14-s + (−0.809 + 0.587i)16-s + (5.23 − 3.80i)17-s + (−0.454 − 1.40i)18-s + (1.11 − 3.44i)19-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.220 − 0.678i)3-s + (0.154 + 0.475i)4-s + (−0.408 + 0.296i)6-s + (−0.0721 − 0.222i)7-s + (0.109 − 0.336i)8-s + (0.396 + 0.288i)9-s + (0.509 + 0.860i)11-s + 0.356·12-s + (0.534 + 0.388i)13-s + (−0.0510 + 0.157i)14-s + (−0.202 + 0.146i)16-s + (1.26 − 0.922i)17-s + (−0.107 − 0.330i)18-s + (0.256 − 0.789i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(4.39177\)
Root analytic conductor: \(2.09565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14089 - 0.632202i\)
\(L(\frac12)\) \(\approx\) \(1.14089 - 0.632202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-1.69 - 2.85i)T \)
good3 \( 1 + (-0.381 + 1.17i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.190 + 0.587i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.92 - 1.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-5.23 + 3.80i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.11 + 3.44i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 + (-1.38 - 4.25i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.73 + 5.34i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.190 + 0.587i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 + (3.11 - 9.59i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.73 + 4.16i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.35 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.61 + 4.08i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.23T + 67T^{2} \)
71 \( 1 + (1.61 - 1.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.472 + 1.45i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5 - 3.63i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.23 - 5.98i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-7.47 - 5.42i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.57583586837593087098802350857, −9.702259361282543488058948923074, −9.032959904270540800332098109175, −7.79886947143502242021820552452, −7.32816312943591065330930594856, −6.41679134912578756343899467374, −4.89562921503912325200188911230, −3.69215664260004750754559127335, −2.30802140684190887493036530957, −1.16326285457883077596620939980, 1.30428425196737489946147590330, 3.27442215457065391043792652179, 4.15022025792735909581657036227, 5.68956763439249199250382304885, 6.20758023619261611541975553191, 7.57066559309663092735657629839, 8.391007945990857418779274943456, 9.117188317937824126051645099847, 10.11983851575812512832974468460, 10.45488803271760838444948290234

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