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Properties

Label 2-224-32.11-c2-0-3
Degree $2$
Conductor $224$
Sign $-0.674 - 0.738i$
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  + (−1.34 − 1.48i)2-s + (1.15 + 2.78i)3-s + (−0.386 + 3.98i)4-s + (−3.15 + 7.61i)5-s + (2.57 − 5.45i)6-s + (1.87 − 1.87i)7-s + (6.41 − 4.77i)8-s + (−0.0699 + 0.0699i)9-s + (15.5 − 5.56i)10-s + (−3.33 + 8.05i)11-s + (−11.5 + 3.51i)12-s + (−2.60 − 6.28i)13-s + (−5.28 − 0.256i)14-s − 24.8·15-s + (−15.7 − 3.07i)16-s + 13.1i·17-s + ⋯
L(s)  = 1  + (−0.672 − 0.740i)2-s + (0.384 + 0.928i)3-s + (−0.0966 + 0.995i)4-s + (−0.631 + 1.52i)5-s + (0.429 − 0.909i)6-s + (0.267 − 0.267i)7-s + (0.802 − 0.597i)8-s + (−0.00777 + 0.00777i)9-s + (1.55 − 0.556i)10-s + (−0.303 + 0.731i)11-s + (−0.961 + 0.293i)12-s + (−0.200 − 0.483i)13-s + (−0.377 − 0.0182i)14-s − 1.65·15-s + (−0.981 − 0.192i)16-s + 0.774i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.674 - 0.738i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.674 - 0.738i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.320597 + 0.726555i\)
\(L(\frac12)\) \(\approx\) \(0.320597 + 0.726555i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 1.48i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (-1.15 - 2.78i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (3.15 - 7.61i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (3.33 - 8.05i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (2.60 + 6.28i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 13.1iT - 289T^{2} \)
19 \( 1 + (26.6 - 11.0i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (7.99 + 7.99i)T + 529iT^{2} \)
29 \( 1 + (5.63 - 2.33i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 39.2iT - 961T^{2} \)
37 \( 1 + (19.0 - 46.0i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (19.5 - 19.5i)T - 1.68e3iT^{2} \)
43 \( 1 + (4.86 - 11.7i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 89.7T + 2.20e3T^{2} \)
53 \( 1 + (68.5 + 28.3i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-104. - 43.1i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-27.7 + 11.4i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-3.80 - 9.18i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-10.5 + 10.5i)T - 5.04e3iT^{2} \)
73 \( 1 + (84.7 - 84.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 33.0T + 6.24e3T^{2} \)
83 \( 1 + (-51.6 + 21.3i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-92.6 - 92.6i)T + 7.92e3iT^{2} \)
97 \( 1 - 100.T + 9.40e3T^{2} \)
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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

−12.08526611818104397941425717502, −11.02378610646541366472548118008, −10.31293453758846013482590553677, −9.969439610981305269964002280172, −8.497052679310165158351544525958, −7.66278678148736315953000916998, −6.61236253835479182000745006979, −4.32854634356518120576766009755, −3.60826089777126285443575686625, −2.37563761218944337184535222582, 0.51021352766324380079783503489, 1.93096757381235316536438241314, 4.49428788579120352113984544263, 5.48919460606891958958223367312, 6.92344502449935347681233240482, 7.80953626710450827627043617629, 8.644351682333287524293562416830, 9.050710939921781533799893178298, 10.63567424698020718506503216738, 11.81392386336801798773005662072

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