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Properties

Label 2-224-28.23-c2-0-8
Degree $2$
Conductor $224$
Sign $0.928 - 0.371i$
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  + (−3.85 + 2.22i)3-s + (2.88 − 4.99i)5-s + (−5.85 + 3.82i)7-s + (5.38 − 9.32i)9-s + (16.6 − 9.60i)11-s − 5.23·13-s + 25.6i·15-s + (14.5 + 25.1i)17-s + (9.49 + 5.48i)19-s + (14.0 − 27.7i)21-s + (11.3 + 6.54i)23-s + (−4.12 − 7.14i)25-s + 7.85i·27-s + 46.5·29-s + (10.8 − 6.24i)31-s + ⋯
L(s)  = 1  + (−1.28 + 0.740i)3-s + (0.576 − 0.998i)5-s + (−0.837 + 0.547i)7-s + (0.598 − 1.03i)9-s + (1.51 − 0.873i)11-s − 0.402·13-s + 1.70i·15-s + (0.855 + 1.48i)17-s + (0.499 + 0.288i)19-s + (0.669 − 1.32i)21-s + (0.492 + 0.284i)23-s + (−0.165 − 0.285i)25-s + 0.290i·27-s + 1.60·29-s + (0.349 − 0.201i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.928 - 0.371i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.07259 + 0.206488i\)
\(L(\frac12)\) \(\approx\) \(1.07259 + 0.206488i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (5.85 - 3.82i)T \)
good3 \( 1 + (3.85 - 2.22i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-2.88 + 4.99i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-16.6 + 9.60i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 5.23T + 169T^{2} \)
17 \( 1 + (-14.5 - 25.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.49 - 5.48i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.3 - 6.54i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 46.5T + 841T^{2} \)
31 \( 1 + (-10.8 + 6.24i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-3.90 + 6.76i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 34.5T + 1.68e3T^{2} \)
43 \( 1 - 22.6iT - 1.84e3T^{2} \)
47 \( 1 + (32.2 + 18.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.6 + 66.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (34.2 - 19.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (23.8 - 41.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-77.1 + 44.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 92.5iT - 5.04e3T^{2} \)
73 \( 1 + (-14.4 - 24.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-37.4 - 21.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 6.88iT - 6.88e3T^{2} \)
89 \( 1 + (57.2 - 99.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 117.T + 9.40e3T^{2} \)
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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

−12.13988967013238163661061350027, −11.19727506216730321812412367993, −10.01957582085373985829120005575, −9.425218738911692892065239121105, −8.434935817755624947748671157766, −6.38961031932232811587249914562, −5.89687219396617259724880358590, −4.90452185348289720922092941338, −3.59923024623062535306256826707, −1.06397037607025677737678880048, 0.993141563742037017109697750920, 2.92591021326245570299227580543, 4.74354916479492949386529349950, 6.14485953834460757718992532177, 6.82329539478382466933219351708, 7.27841156067427902470260432528, 9.450934812161793445590323962268, 10.07712100063949030773916006562, 11.14193844024579886503279097550, 12.01795118413892674232314758402

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