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Properties

Label 2-416-52.3-c2-0-18
Degree $2$
Conductor $416$
Sign $-0.451 + 0.892i$
Analytic cond. $11.3351$
Root an. cond. $3.36677$
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  + (−3.45 + 1.99i)3-s + 6.36·5-s + (−5.32 − 3.07i)7-s + (3.46 − 6.00i)9-s + (−4.55 + 2.63i)11-s + (−0.635 − 12.9i)13-s + (−22.0 + 12.7i)15-s + (−16.0 + 27.8i)17-s + (−4.23 − 2.44i)19-s + 24.5·21-s + (8.99 − 5.19i)23-s + 15.5·25-s − 8.25i·27-s + (−18.6 − 32.3i)29-s − 39.4i·31-s + ⋯
L(s)  = 1  + (−1.15 + 0.665i)3-s + 1.27·5-s + (−0.760 − 0.439i)7-s + (0.385 − 0.666i)9-s + (−0.414 + 0.239i)11-s + (−0.0488 − 0.998i)13-s + (−1.46 + 0.847i)15-s + (−0.944 + 1.63i)17-s + (−0.223 − 0.128i)19-s + 1.16·21-s + (0.391 − 0.225i)23-s + 0.622·25-s − 0.305i·27-s + (−0.644 − 1.11i)29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(11.3351\)
Root analytic conductor: \(3.36677\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.198792 - 0.323325i\)
\(L(\frac12)\) \(\approx\) \(0.198792 - 0.323325i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.635 + 12.9i)T \)
good3 \( 1 + (3.45 - 1.99i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 - 6.36T + 25T^{2} \)
7 \( 1 + (5.32 + 3.07i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.55 - 2.63i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (16.0 - 27.8i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (4.23 + 2.44i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.99 + 5.19i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (18.6 + 32.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 39.4iT - 961T^{2} \)
37 \( 1 + (16.8 + 29.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (15.9 + 27.6i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (7.50 + 4.33i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 69.4iT - 2.20e3T^{2} \)
53 \( 1 + 19.7T + 2.80e3T^{2} \)
59 \( 1 + (80.1 + 46.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-29.2 + 50.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-113. + 65.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (40.2 + 23.2i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 28.9T + 5.32e3T^{2} \)
79 \( 1 - 73.0iT - 6.24e3T^{2} \)
83 \( 1 - 6.44iT - 6.88e3T^{2} \)
89 \( 1 + (41.4 + 71.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-28.4 + 49.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.56281675520161501444196270614, −10.06261194710503059518534520283, −9.284858116968894961187913046266, −7.907158715942576164103767388150, −6.40443921414720667604683315323, −5.99107478571494714553053864902, −5.05577243616954062218791693962, −3.87403610868585358027576018812, −2.19476427243488070828132163527, −0.17898775053534477764840027404, 1.57703155684820358377242280462, 2.88656190059502215311821061700, 4.95812105430983240014568723333, 5.63938132088396101288558290315, 6.66277378633139412120882097020, 6.96761042299420062585504633657, 8.844572953325426417546234140589, 9.471389290085343568163842889461, 10.47149483243446013387436789823, 11.41544496554546230814868483708

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