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Properties

Label 2-224-224.13-c2-0-29
Degree $2$
Conductor $224$
Sign $-0.854 + 0.520i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 − 1.49i)2-s + (−2.62 − 1.08i)3-s + (−0.450 + 3.97i)4-s + (2.38 + 5.76i)5-s + (1.87 + 5.37i)6-s + (−3.62 + 5.98i)7-s + (6.52 − 4.62i)8-s + (−0.638 − 0.638i)9-s + (5.42 − 11.2i)10-s + (−4.95 − 11.9i)11-s + (5.51 − 9.95i)12-s + (6.06 − 14.6i)13-s + (13.7 − 2.55i)14-s − 17.7i·15-s + (−15.5 − 3.58i)16-s + 8.22·17-s + ⋯
L(s)  = 1  + (−0.666 − 0.745i)2-s + (−0.876 − 0.362i)3-s + (−0.112 + 0.993i)4-s + (0.477 + 1.15i)5-s + (0.312 + 0.895i)6-s + (−0.518 + 0.855i)7-s + (0.816 − 0.577i)8-s + (−0.0709 − 0.0709i)9-s + (0.542 − 1.12i)10-s + (−0.450 − 1.08i)11-s + (0.459 − 0.829i)12-s + (0.466 − 1.12i)13-s + (0.983 − 0.182i)14-s − 1.18i·15-s + (−0.974 − 0.223i)16-s + 0.484·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.854 + 0.520i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.854 + 0.520i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.101872 - 0.363127i\)
\(L(\frac12)\) \(\approx\) \(0.101872 - 0.363127i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 1.49i)T \)
7 \( 1 + (3.62 - 5.98i)T \)
good3 \( 1 + (2.62 + 1.08i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-2.38 - 5.76i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (4.95 + 11.9i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-6.06 + 14.6i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 8.22T + 289T^{2} \)
19 \( 1 + (8.30 - 20.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (26.4 + 26.4i)T + 529iT^{2} \)
29 \( 1 + (-7.68 + 18.5i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 59.4iT - 961T^{2} \)
37 \( 1 + (20.1 - 8.33i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-24.3 + 24.3i)T - 1.68e3iT^{2} \)
43 \( 1 + (14.7 + 35.5i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 5.31T + 2.20e3T^{2} \)
53 \( 1 + (-15.0 - 36.4i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-6.55 - 15.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (86.2 + 35.7i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-3.59 + 8.67i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-22.3 + 22.3i)T - 5.04e3iT^{2} \)
73 \( 1 + (92.1 - 92.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 125. iT - 6.24e3T^{2} \)
83 \( 1 + (-9.80 + 23.6i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (22.0 + 22.0i)T + 7.92e3iT^{2} \)
97 \( 1 + 126. iT - 9.40e3T^{2} \)
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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

−11.55083301099138227150367365401, −10.55571850656787495028795628223, −10.14850799806926728801672644990, −8.723752994769421752154160712350, −7.77056869556363567973913960857, −6.18493520913655323434406937353, −5.90139496882801801117563456851, −3.45544781883089420419762091274, −2.44819130962820711652253897701, −0.29297145795996161026009214963, 1.45487215046419501328045702234, 4.50053079278791400605442827073, 5.17209813765128343509737909226, 6.32582505971469723640750205176, 7.30167009904976040994138531905, 8.604642645505276682109272612818, 9.552390083329834311238654615866, 10.24109449731923531748663218228, 11.19421231566478727966175364362, 12.40909801574627557593793069427

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