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Properties

Label 2-704-44.43-c3-0-40
Degree $2$
Conductor $704$
Sign $-0.538 + 0.842i$
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  − 2.43i·3-s − 17.0·5-s + 13.3·7-s + 21.0·9-s + (−30.7 − 19.6i)11-s + 66.3i·13-s + 41.4i·15-s − 10.3i·17-s + 86.7·19-s − 32.3i·21-s − 134. i·23-s + 165.·25-s − 116. i·27-s + 69.9i·29-s + 79.2i·31-s + ⋯
L(s)  = 1  − 0.468i·3-s − 1.52·5-s + 0.718·7-s + 0.780·9-s + (−0.842 − 0.538i)11-s + 1.41i·13-s + 0.713i·15-s − 0.148i·17-s + 1.04·19-s − 0.336i·21-s − 1.21i·23-s + 1.32·25-s − 0.833i·27-s + 0.448i·29-s + 0.459i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $-0.538 + 0.842i$
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: \((2,\ 704,\ (\ :3/2),\ -0.538 + 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.006082709\)
\(L(\frac12)\) \(\approx\) \(1.006082709\)
\(L(\frac{5}{2})\) 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 \( 1 + (30.7 + 19.6i)T \)
good3 \( 1 + 2.43iT - 27T^{2} \)
5 \( 1 + 17.0T + 125T^{2} \)
7 \( 1 - 13.3T + 343T^{2} \)
13 \( 1 - 66.3iT - 2.19e3T^{2} \)
17 \( 1 + 10.3iT - 4.91e3T^{2} \)
19 \( 1 - 86.7T + 6.85e3T^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 - 69.9iT - 2.43e4T^{2} \)
31 \( 1 - 79.2iT - 2.97e4T^{2} \)
37 \( 1 - 111.T + 5.06e4T^{2} \)
41 \( 1 + 261. iT - 6.89e4T^{2} \)
43 \( 1 - 128.T + 7.95e4T^{2} \)
47 \( 1 - 422. iT - 1.03e5T^{2} \)
53 \( 1 + 727.T + 1.48e5T^{2} \)
59 \( 1 + 573. iT - 2.05e5T^{2} \)
61 \( 1 + 889. iT - 2.26e5T^{2} \)
67 \( 1 + 137. iT - 3.00e5T^{2} \)
71 \( 1 - 35.6iT - 3.57e5T^{2} \)
73 \( 1 + 845. iT - 3.89e5T^{2} \)
79 \( 1 + 603.T + 4.93e5T^{2} \)
83 \( 1 + 1.07e3T + 5.71e5T^{2} \)
89 \( 1 + 532.T + 7.04e5T^{2} \)
97 \( 1 + 371.T + 9.12e5T^{2} \)
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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

−9.717639417181411391363859965762, −8.626863272474772399918590576790, −7.87002044327001079960305264699, −7.34309426290519193317285981916, −6.44674888058049460212518209127, −4.89880766437497964726141673538, −4.30565168918242874325845871323, −3.12479026712083040679502168705, −1.64022358522591915610003024480, −0.32807949138591731956054618186, 1.14162841514900329253475179444, 2.94316567923285852854678772988, 3.92969193959154399555273965794, 4.75799317601799152319875785118, 5.57078002129068740959936027582, 7.34317354405751054389567657131, 7.65318573186951574195336598490, 8.342646876769460105003922940961, 9.669572792312280793778260733617, 10.33387051771875004385989646718

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