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Properties

Label 2-224-224.13-c2-0-9
Degree $2$
Conductor $224$
Sign $0.357 - 0.933i$
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  + (−1.84 + 0.781i)2-s + (−3.58 − 1.48i)3-s + (2.77 − 2.87i)4-s + (3.04 + 7.34i)5-s + (7.75 − 0.0689i)6-s + (−1.24 − 6.88i)7-s + (−2.86 + 7.47i)8-s + (4.27 + 4.27i)9-s + (−11.3 − 11.1i)10-s + (−5.50 − 13.2i)11-s + (−14.2 + 6.19i)12-s + (−2.01 + 4.86i)13-s + (7.68 + 11.7i)14-s − 30.8i·15-s + (−0.568 − 15.9i)16-s + 22.4·17-s + ⋯
L(s)  = 1  + (−0.920 + 0.390i)2-s + (−1.19 − 0.494i)3-s + (0.694 − 0.719i)4-s + (0.608 + 1.46i)5-s + (1.29 − 0.0114i)6-s + (−0.178 − 0.984i)7-s + (−0.357 + 0.933i)8-s + (0.475 + 0.475i)9-s + (−1.13 − 1.11i)10-s + (−0.500 − 1.20i)11-s + (−1.18 + 0.516i)12-s + (−0.155 + 0.374i)13-s + (0.548 + 0.836i)14-s − 2.05i·15-s + (−0.0355 − 0.999i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.357 - 0.933i$
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: \((2,\ 224,\ (\ :1),\ 0.357 - 0.933i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.504460 + 0.347088i\)
\(L(\frac12)\) \(\approx\) \(0.504460 + 0.347088i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.84 - 0.781i)T \)
7 \( 1 + (1.24 + 6.88i)T \)
good3 \( 1 + (3.58 + 1.48i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-3.04 - 7.34i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (5.50 + 13.2i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (2.01 - 4.86i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 22.4T + 289T^{2} \)
19 \( 1 + (11.7 - 28.3i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-29.8 - 29.8i)T + 529iT^{2} \)
29 \( 1 + (0.784 - 1.89i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 6.89iT - 961T^{2} \)
37 \( 1 + (-40.9 + 16.9i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (11.9 - 11.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-23.7 - 57.2i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 11.6T + 2.20e3T^{2} \)
53 \( 1 + (-20.4 - 49.4i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-18.1 - 43.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-13.0 - 5.42i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (41.7 - 100. i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-36.1 + 36.1i)T - 5.04e3iT^{2} \)
73 \( 1 + (-36.3 + 36.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 20.3iT - 6.24e3T^{2} \)
83 \( 1 + (-26.5 + 63.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-24.1 - 24.1i)T + 7.92e3iT^{2} \)
97 \( 1 + 24.9iT - 9.40e3T^{2} \)
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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

−11.72712687817960439961437117542, −10.94024774840715024372410357858, −10.49508918433682684246741653074, −9.594953588361418027504793822193, −7.83213489436002532710698773214, −7.12237815457608345513978707567, −6.14748197137848677961739266351, −5.67670559151671343515979785360, −3.14579656238882017388742172715, −1.16912270320393332531154344397, 0.61591258763418532504696619907, 2.39482421728339646072317731844, 4.75813590577771783153778413099, 5.39204101421299248047667392569, 6.67259302246004359561908790576, 8.221249696085353359715864641136, 9.150526303962062971975451307797, 9.862887040385452245874117735237, 10.73095052699141169221669444435, 11.85927187160792572620721388331

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