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Properties

Label 2-408-17.2-c1-0-4
Degree $2$
Conductor $408$
Sign $0.804 - 0.593i$
Analytic cond. $3.25789$
Root an. cond. $1.80496$
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  + (0.923 − 0.382i)3-s + (0.431 + 1.04i)5-s + (−1.16 + 2.80i)7-s + (0.707 − 0.707i)9-s + (3.84 + 1.59i)11-s − 1.86i·13-s + (0.797 + 0.797i)15-s + (−1.96 + 3.62i)17-s + (5.77 + 5.77i)19-s + 3.04i·21-s + (−6.03 − 2.49i)23-s + (2.63 − 2.63i)25-s + (0.382 − 0.923i)27-s + (−2.72 − 6.58i)29-s + (7.71 − 3.19i)31-s + ⋯
L(s)  = 1  + (0.533 − 0.220i)3-s + (0.193 + 0.466i)5-s + (−0.439 + 1.06i)7-s + (0.235 − 0.235i)9-s + (1.15 + 0.480i)11-s − 0.516i·13-s + (0.206 + 0.206i)15-s + (−0.475 + 0.879i)17-s + (1.32 + 1.32i)19-s + 0.663i·21-s + (−1.25 − 0.521i)23-s + (0.527 − 0.527i)25-s + (0.0736 − 0.177i)27-s + (−0.506 − 1.22i)29-s + (1.38 − 0.574i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(408\)    =    \(2^{3} \cdot 3 \cdot 17\)
Sign: $0.804 - 0.593i$
Analytic conductor: \(3.25789\)
Root analytic conductor: \(1.80496\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{408} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 408,\ (\ :1/2),\ 0.804 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59334 + 0.524105i\)
\(L(\frac12)\) \(\approx\) \(1.59334 + 0.524105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (1.96 - 3.62i)T \)
good5 \( 1 + (-0.431 - 1.04i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.16 - 2.80i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-3.84 - 1.59i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.86iT - 13T^{2} \)
19 \( 1 + (-5.77 - 5.77i)T + 19iT^{2} \)
23 \( 1 + (6.03 + 2.49i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.72 + 6.58i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-7.71 + 3.19i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.323 + 0.134i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.40 + 3.39i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (6.71 - 6.71i)T - 43iT^{2} \)
47 \( 1 - 4.96iT - 47T^{2} \)
53 \( 1 + (7.13 + 7.13i)T + 53iT^{2} \)
59 \( 1 + (-0.0213 + 0.0213i)T - 59iT^{2} \)
61 \( 1 + (-1.66 + 4.02i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 5.70T + 67T^{2} \)
71 \( 1 + (3.98 - 1.65i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.25 + 7.85i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.15 + 3.37i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 - 1.83iT - 89T^{2} \)
97 \( 1 + (4.76 + 11.5i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.57352285935510076460060730452, −10.06536238959211116875992160689, −9.670708413606491437955524127063, −8.534780552203715398486226040234, −7.78534009130852819888064025334, −6.41047784206628343331603960241, −5.95844657129974206073255918754, −4.23948297005497249600700114071, −3.06747024927656244847667188088, −1.87766364272804359388755197878, 1.20133531873565416401679627030, 3.09955620857960024884494148621, 4.11239918660868690850462029851, 5.16034456560499637107244232927, 6.70018569656499360936694092234, 7.27362698984563076262503675033, 8.655484761648791705151652452431, 9.318976820928644399596419892796, 10.00866707180091653223038450325, 11.21364075606127268609965094376

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