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Properties

Label 2-416-104.11-c1-0-11
Degree $2$
Conductor $416$
Sign $-0.735 - 0.677i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (−1.11 − 1.92i)3-s + (0.0693 − 0.0693i)5-s + (−3.36 + 0.901i)7-s + (−0.979 + 1.69i)9-s + (0.305 + 0.0819i)11-s + (−3.13 + 1.78i)13-s + (−0.210 − 0.0565i)15-s + (−5.48 − 3.16i)17-s + (0.397 − 0.106i)19-s + (5.48 + 5.48i)21-s + (3.68 + 6.38i)23-s + 4.99i·25-s − 2.32·27-s + (−3.16 + 1.82i)29-s + (−1.12 + 1.12i)31-s + ⋯
L(s)  = 1  + (−0.642 − 1.11i)3-s + (0.0310 − 0.0310i)5-s + (−1.27 + 0.340i)7-s + (−0.326 + 0.565i)9-s + (0.0921 + 0.0247i)11-s + (−0.868 + 0.495i)13-s + (−0.0544 − 0.0145i)15-s + (−1.33 − 0.768i)17-s + (0.0911 − 0.0244i)19-s + (1.19 + 1.19i)21-s + (0.768 + 1.33i)23-s + 0.998i·25-s − 0.446·27-s + (−0.588 + 0.339i)29-s + (−0.201 + 0.201i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0166899 + 0.0427574i\)
\(L(\frac12)\) \(\approx\) \(0.0166899 + 0.0427574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (3.13 - 1.78i)T \)
good3 \( 1 + (1.11 + 1.92i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.0693 + 0.0693i)T - 5iT^{2} \)
7 \( 1 + (3.36 - 0.901i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.305 - 0.0819i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (5.48 + 3.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.397 + 0.106i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.68 - 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.12 - 1.12i)T - 31iT^{2} \)
37 \( 1 + (-0.804 + 3.00i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.85 + 6.91i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.88 + 2.82i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.15 + 6.15i)T + 47iT^{2} \)
53 \( 1 - 3.89iT - 53T^{2} \)
59 \( 1 + (-2.51 - 9.38i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (11.9 + 6.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 11.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.81 + 6.78i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.49 + 1.49i)T - 73iT^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (2.53 + 2.53i)T + 83iT^{2} \)
89 \( 1 + (-0.110 - 0.0296i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.28 + 1.68i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.90413344058827020232826635867, −9.471131725698682293257869361388, −9.120157491401255198346623103186, −7.34575926149391197019399540321, −7.00261361513033637645140801573, −6.06065688122737281705935328249, −5.04854973183695569466040269709, −3.36570430643453266443203606513, −1.94131753314559723368017901025, −0.02979527907365737236145918820, 2.76511263215462049340564738413, 4.07319329257073920655319450545, 4.85056821280595744400649037978, 6.12348335037236385888045537659, 6.81535522783244125645748530173, 8.244653512599268273693937680078, 9.435014892018870551809125194259, 10.02014088901205016153959024391, 10.69257906171311018407543072522, 11.51523840311068410902391198781

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