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Properties

Label 2-224-32.29-c1-0-9
Degree $2$
Conductor $224$
Sign $0.967 + 0.254i$
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  + (−0.420 − 1.35i)2-s + (2.28 + 0.946i)3-s + (−1.64 + 1.13i)4-s + (0.446 + 1.07i)5-s + (0.316 − 3.48i)6-s + (−0.707 + 0.707i)7-s + (2.22 + 1.74i)8-s + (2.20 + 2.20i)9-s + (1.26 − 1.05i)10-s + (2.89 − 1.19i)11-s + (−4.83 + 1.03i)12-s + (1.10 − 2.65i)13-s + (1.25 + 0.657i)14-s + 2.88i·15-s + (1.41 − 3.73i)16-s + 3.12i·17-s + ⋯
L(s)  = 1  + (−0.297 − 0.954i)2-s + (1.31 + 0.546i)3-s + (−0.823 + 0.567i)4-s + (0.199 + 0.481i)5-s + (0.129 − 1.42i)6-s + (−0.267 + 0.267i)7-s + (0.786 + 0.616i)8-s + (0.734 + 0.734i)9-s + (0.400 − 0.333i)10-s + (0.871 − 0.361i)11-s + (−1.39 + 0.299i)12-s + (0.305 − 0.737i)13-s + (0.334 + 0.175i)14-s + 0.744i·15-s + (0.354 − 0.934i)16-s + 0.758i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48180 - 0.191665i\)
\(L(\frac12)\) \(\approx\) \(1.48180 - 0.191665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.420 + 1.35i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-2.28 - 0.946i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.446 - 1.07i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (-2.89 + 1.19i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.10 + 2.65i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 - 3.12iT - 17T^{2} \)
19 \( 1 + (1.63 - 3.95i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.37 + 1.37i)T + 23iT^{2} \)
29 \( 1 + (8.15 + 3.37i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 2.07T + 31T^{2} \)
37 \( 1 + (2.47 + 5.96i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.68 + 5.68i)T + 41iT^{2} \)
43 \( 1 + (-5.63 + 2.33i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.41iT - 47T^{2} \)
53 \( 1 + (12.4 - 5.16i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.20 - 5.32i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-4.89 - 2.02i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-8.29 - 3.43i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-6.35 + 6.35i)T - 71iT^{2} \)
73 \( 1 + (10.6 + 10.6i)T + 73iT^{2} \)
79 \( 1 - 5.81iT - 79T^{2} \)
83 \( 1 + (1.91 - 4.62i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-5.79 + 5.79i)T - 89iT^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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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

−12.27137396885066904687880781729, −10.94557748636939011998116246031, −10.22830564268049249689419073921, −9.292013894446044270536117356362, −8.628657093929688557906532704846, −7.71911397627063791466536126251, −5.95373136539693982063021914379, −4.02766970452384769424172380369, −3.35250040497967046038613686707, −2.09967920408832612637111031638, 1.63372582738785931005600295559, 3.64169716848360266283034707439, 4.99475732920135165807201801437, 6.64562693086589951070332328279, 7.24485934754130908292217007268, 8.402341960555268608076160067882, 9.164878183352105001720478028976, 9.640268155984212018461125683629, 11.32471952101022801504753017670, 12.88709313463125116171581893685

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