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Properties

Label 2-224-28.11-c2-0-1
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: $\chi_{224} (95, \cdot )$
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

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

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