Properties

Label 2-224-8.5-c1-0-5
Degree $2$
Conductor $224$
Sign $-0.552 + 0.833i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.02i·3-s − 1.69i·5-s + 7-s − 6.12·9-s + 1.32i·11-s + 1.69i·13-s − 5.12·15-s + 2·17-s − 3.02i·19-s − 3.02i·21-s − 5.12·23-s + 2.12·25-s + 9.43i·27-s − 6.04i·29-s + 10.2·31-s + ⋯
L(s)  = 1  − 1.74i·3-s − 0.758i·5-s + 0.377·7-s − 2.04·9-s + 0.399i·11-s + 0.470i·13-s − 1.32·15-s + 0.485·17-s − 0.692i·19-s − 0.659i·21-s − 1.06·23-s + 0.424·25-s + 1.81i·27-s − 1.12i·29-s + 1.84·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.552 + 0.833i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.552 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562052 - 1.04626i\)
\(L(\frac12)\) \(\approx\) \(0.562052 - 1.04626i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + 3.02iT - 3T^{2} \)
5 \( 1 + 1.69iT - 5T^{2} \)
11 \( 1 - 1.32iT - 11T^{2} \)
13 \( 1 - 1.69iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 3.02iT - 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 + 6.04iT - 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 6.04iT - 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 2.64iT - 53T^{2} \)
59 \( 1 - 0.371iT - 59T^{2} \)
61 \( 1 - 1.69iT - 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 5.66iT - 83T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−12.02760016209081869353038293576, −11.49159043456510829847509753241, −9.877488026660716109241297158765, −8.570551349838598725734842522995, −7.943827082700113966125576336216, −6.91466619904657404457790069855, −5.94554954478163185201880575076, −4.59296494557860739664957514443, −2.44699884858925101646363558542, −1.10029157220730630631749103386, 2.94534133514974689881576895124, 3.95544329584768073844385342451, 5.14750988930028947081318611008, 6.17370794285566875514273294492, 7.83481488499514051033969834975, 8.835398432905057936421975266384, 9.968688227871516142777820088516, 10.51698832503969239090846631852, 11.24995763869216456485301347960, 12.32239993799148269669245319068

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