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Properties

Label 2-224-32.3-c2-0-6
Degree $2$
Conductor $224$
Sign $-0.306 - 0.951i$
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  + (−1.00 + 1.72i)2-s + (−0.343 + 0.830i)3-s + (−1.97 − 3.47i)4-s + (−1.68 − 4.07i)5-s + (−1.08 − 1.42i)6-s + (−1.87 − 1.87i)7-s + (7.99 + 0.0754i)8-s + (5.79 + 5.79i)9-s + (8.74 + 1.17i)10-s + (1.83 + 4.42i)11-s + (3.56 − 0.446i)12-s + (−5.16 + 12.4i)13-s + (5.11 − 1.35i)14-s + 3.96·15-s + (−8.17 + 13.7i)16-s + 25.5i·17-s + ⋯
L(s)  = 1  + (−0.502 + 0.864i)2-s + (−0.114 + 0.276i)3-s + (−0.494 − 0.869i)4-s + (−0.337 − 0.815i)5-s + (−0.181 − 0.238i)6-s + (−0.267 − 0.267i)7-s + (0.999 + 0.00943i)8-s + (0.643 + 0.643i)9-s + (0.874 + 0.117i)10-s + (0.166 + 0.402i)11-s + (0.297 − 0.0372i)12-s + (−0.397 + 0.958i)13-s + (0.365 − 0.0966i)14-s + 0.264·15-s + (−0.510 + 0.859i)16-s + 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.306 - 0.951i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.306 - 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.554153 + 0.760623i\)
\(L(\frac12)\) \(\approx\) \(0.554153 + 0.760623i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.00 - 1.72i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (0.343 - 0.830i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (1.68 + 4.07i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-1.83 - 4.42i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (5.16 - 12.4i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 25.5iT - 289T^{2} \)
19 \( 1 + (-20.0 - 8.29i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-19.0 + 19.0i)T - 529iT^{2} \)
29 \( 1 + (28.8 + 11.9i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 20.2iT - 961T^{2} \)
37 \( 1 + (-20.6 - 49.7i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-48.4 - 48.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (1.98 + 4.80i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 0.301T + 2.20e3T^{2} \)
53 \( 1 + (57.5 - 23.8i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-10.1 + 4.18i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-41.4 - 17.1i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-20.6 + 49.8i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (61.3 + 61.3i)T + 5.04e3iT^{2} \)
73 \( 1 + (-41.1 - 41.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 68.4T + 6.24e3T^{2} \)
83 \( 1 + (105. + 43.7i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-48.8 + 48.8i)T - 7.92e3iT^{2} \)
97 \( 1 + 140.T + 9.40e3T^{2} \)
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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

−12.50052092373987045923725736819, −11.11992228679861328893858579735, −10.08223291277630275832676525397, −9.355708313338311643740551753389, −8.281571310056606394904526807656, −7.40410587001515296178589191297, −6.31833305118617793860549743203, −4.91662742439320055960589760297, −4.22093091056358563241012139175, −1.42567693098736823053473029966, 0.70190009597951723264134841132, 2.74331056796888542935867375505, 3.64364978227610726603125623574, 5.37113232703055295570628362185, 7.14047230204638793989613108306, 7.52649044704755506164601834870, 9.200402005050949253352628979998, 9.690100293817755003649556146754, 11.00409644733020379339838569995, 11.51968603207381760932988844701

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