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Properties

Label 2-704-44.43-c3-0-69
Degree $2$
Conductor $704$
Sign $-0.00559 - 0.999i$
Analytic cond. $41.5373$
Root an. cond. $6.44494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.41i·3-s + 17.7·5-s − 30.3·7-s − 61.6·9-s + (−36.4 + 0.203i)11-s − 73.3i·13-s − 167. i·15-s + 33.5i·17-s − 3.79·19-s + 286. i·21-s + 39.3i·23-s + 191.·25-s + 326. i·27-s + 80.3i·29-s − 132. i·31-s + ⋯
L(s)  = 1  − 1.81i·3-s + 1.59·5-s − 1.64·7-s − 2.28·9-s + (−0.999 + 0.00559i)11-s − 1.56i·13-s − 2.88i·15-s + 0.479i·17-s − 0.0457·19-s + 2.97i·21-s + 0.356i·23-s + 1.53·25-s + 2.32i·27-s + 0.514i·29-s − 0.765i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00559 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00559 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.00559 - 0.999i$
Analytic conductor: \(41.5373\)
Root analytic conductor: \(6.44494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :3/2),\ -0.00559 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3372719621\)
\(L(\frac12)\) \(\approx\) \(0.3372719621\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (36.4 - 0.203i)T \)
good3 \( 1 + 9.41iT - 27T^{2} \)
5 \( 1 - 17.7T + 125T^{2} \)
7 \( 1 + 30.3T + 343T^{2} \)
13 \( 1 + 73.3iT - 2.19e3T^{2} \)
17 \( 1 - 33.5iT - 4.91e3T^{2} \)
19 \( 1 + 3.79T + 6.85e3T^{2} \)
23 \( 1 - 39.3iT - 1.21e4T^{2} \)
29 \( 1 - 80.3iT - 2.43e4T^{2} \)
31 \( 1 + 132. iT - 2.97e4T^{2} \)
37 \( 1 - 177.T + 5.06e4T^{2} \)
41 \( 1 - 155. iT - 6.89e4T^{2} \)
43 \( 1 + 49.0T + 7.95e4T^{2} \)
47 \( 1 - 363. iT - 1.03e5T^{2} \)
53 \( 1 + 313.T + 1.48e5T^{2} \)
59 \( 1 - 31.5iT - 2.05e5T^{2} \)
61 \( 1 + 204. iT - 2.26e5T^{2} \)
67 \( 1 - 638. iT - 3.00e5T^{2} \)
71 \( 1 + 712. iT - 3.57e5T^{2} \)
73 \( 1 - 747. iT - 3.89e5T^{2} \)
79 \( 1 + 722.T + 4.93e5T^{2} \)
83 \( 1 + 6.82T + 5.71e5T^{2} \)
89 \( 1 + 1.16e3T + 7.04e5T^{2} \)
97 \( 1 + 855.T + 9.12e5T^{2} \)
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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

−9.498214370943634497301930480951, −8.355534277959464011728141179689, −7.53512935196894053269032316469, −6.58455748801203659703397550170, −5.93965819198819338660919211236, −5.50928796824662051658893455460, −3.00706882076931298078800680452, −2.55352570210430572453925181930, −1.27178983975968045192267030694, −0.088735939971774495190262529449, 2.34630136949621956119987723359, 3.17154269054928032326509891252, 4.33049804620008595349096760050, 5.28721729345449353204330671446, 6.04292972992715083655535677014, 6.83460653481022627542743668620, 8.685506464689822769976081483394, 9.381725584008299436046015591881, 9.781000488723839833865838342479, 10.29743631563654385910472198511

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