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Properties

Label 2-224-224.69-c2-0-18
Degree $2$
Conductor $224$
Sign $0.916 - 0.400i$
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  + (−1.46 − 1.36i)2-s + (−1.65 + 0.686i)3-s + (0.300 + 3.98i)4-s + (0.205 − 0.495i)5-s + (3.36 + 1.24i)6-s + (−0.193 − 6.99i)7-s + (4.98 − 6.25i)8-s + (−4.08 + 4.08i)9-s + (−0.974 + 0.447i)10-s + (−3.44 + 8.32i)11-s + (−3.23 − 6.40i)12-s + (−1.31 − 3.17i)13-s + (−9.23 + 10.5i)14-s + 0.962i·15-s + (−15.8 + 2.39i)16-s + 28.1·17-s + ⋯
L(s)  = 1  + (−0.733 − 0.680i)2-s + (−0.552 + 0.228i)3-s + (0.0751 + 0.997i)4-s + (0.0410 − 0.0990i)5-s + (0.560 + 0.207i)6-s + (−0.0276 − 0.999i)7-s + (0.623 − 0.782i)8-s + (−0.454 + 0.454i)9-s + (−0.0974 + 0.0447i)10-s + (−0.313 + 0.757i)11-s + (−0.269 − 0.533i)12-s + (−0.101 − 0.243i)13-s + (−0.659 + 0.751i)14-s + 0.0641i·15-s + (−0.988 + 0.149i)16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.744063 + 0.155477i\)
\(L(\frac12)\) \(\approx\) \(0.744063 + 0.155477i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.46 + 1.36i)T \)
7 \( 1 + (0.193 + 6.99i)T \)
good3 \( 1 + (1.65 - 0.686i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-0.205 + 0.495i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (3.44 - 8.32i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (1.31 + 3.17i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 28.1T + 289T^{2} \)
19 \( 1 + (-12.1 - 29.2i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (11.0 - 11.0i)T - 529iT^{2} \)
29 \( 1 + (-6.55 - 15.8i)T + (-594. + 594. i)T^{2} \)
31 \( 1 + 31.5iT - 961T^{2} \)
37 \( 1 + (-39.9 - 16.5i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (-53.7 - 53.7i)T + 1.68e3iT^{2} \)
43 \( 1 + (-8.17 + 19.7i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 30.9T + 2.20e3T^{2} \)
53 \( 1 + (38.3 - 92.4i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (36.8 - 89.0i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-80.8 + 33.4i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (4.47 + 10.7i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (63.6 + 63.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (35.2 + 35.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (-30.0 - 72.5i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (23.6 - 23.6i)T - 7.92e3iT^{2} \)
97 \( 1 + 115. iT - 9.40e3T^{2} \)
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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

−11.92152186069314253657962809772, −10.96144174850314727201472906307, −10.15740584250272240440979742383, −9.616253465089523233045281483081, −7.891192236138098809756705147131, −7.58268191419429410483215121795, −5.83882084607306153491043971405, −4.48132168059985281737463278620, −3.14543409691064473144598828312, −1.24153087912994893504295075585, 0.66662946092202651259955506463, 2.78364706974234337875268784303, 5.14624696956469660691536014757, 5.89203752908280390288542733117, 6.78438577861392188219246880434, 8.066609858103739322757049651409, 8.922345804880036098213347804664, 9.818492727708503927983902273579, 11.03910909740489098516080295203, 11.74375701373257195644017045100

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