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Properties

Label 2-224-32.5-c1-0-14
Degree $2$
Conductor $224$
Sign $0.963 - 0.268i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
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  + (1.35 + 0.416i)2-s + (−0.244 − 0.590i)3-s + (1.65 + 1.12i)4-s + (−0.220 − 0.0913i)5-s + (−0.0844 − 0.899i)6-s + (0.707 + 0.707i)7-s + (1.76 + 2.21i)8-s + (1.83 − 1.83i)9-s + (−0.259 − 0.215i)10-s + (−0.352 + 0.851i)11-s + (0.260 − 1.25i)12-s + (−1.31 + 0.545i)13-s + (0.660 + 1.25i)14-s + 0.152i·15-s + (1.46 + 3.72i)16-s − 3.60i·17-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.141 − 0.340i)3-s + (0.826 + 0.563i)4-s + (−0.0986 − 0.0408i)5-s + (−0.0344 − 0.367i)6-s + (0.267 + 0.267i)7-s + (0.623 + 0.781i)8-s + (0.610 − 0.610i)9-s + (−0.0821 − 0.0680i)10-s + (−0.106 + 0.256i)11-s + (0.0752 − 0.361i)12-s + (−0.365 + 0.151i)13-s + (0.176 + 0.334i)14-s + 0.0393i·15-s + (0.365 + 0.930i)16-s − 0.874i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.963 - 0.268i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.963 - 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01176 + 0.274746i\)
\(L(\frac12)\) \(\approx\) \(2.01176 + 0.274746i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.416i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.244 + 0.590i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.220 + 0.0913i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.352 - 0.851i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.31 - 0.545i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 3.60iT - 17T^{2} \)
19 \( 1 + (3.74 - 1.55i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.802 - 0.802i)T - 23iT^{2} \)
29 \( 1 + (-0.583 - 1.40i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (2.19 + 0.910i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.11 + 4.11i)T - 41iT^{2} \)
43 \( 1 + (2.07 - 5.01i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 0.582iT - 47T^{2} \)
53 \( 1 + (-1.75 + 4.23i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (7.70 + 3.19i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.59 - 11.0i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.28 + 7.92i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-10.8 - 10.8i)T + 71iT^{2} \)
73 \( 1 + (-9.19 + 9.19i)T - 73iT^{2} \)
79 \( 1 + 5.19iT - 79T^{2} \)
83 \( 1 + (-9.73 + 4.03i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.11 - 7.11i)T + 89iT^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−12.33900822016598049338605674513, −11.76827995096902210031748365209, −10.62033954122241356963336672360, −9.338374010993986115330071882629, −7.963349491414491729402884347008, −7.08167428759787283357379099534, −6.12890791451371851088770287969, −4.92157451474387310511121395082, −3.80253688787973389638418188913, −2.12754267525031540590401772199, 1.99237847481692191168527346196, 3.71740997542298460032190757018, 4.67767210462556832887335887388, 5.74147636254752211161488733196, 7.00124183753048856493982514406, 8.041428058190397849579081672509, 9.650984841815102216790557545525, 10.64974138491144678528213421348, 11.12113616659095776415195394727, 12.36417639337934564793192241152

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