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Properties

Label 2-416-52.47-c1-0-13
Degree $2$
Conductor $416$
Sign $-0.971 - 0.238i$
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  − 2.89i·3-s + (−2.84 − 2.84i)5-s + (−0.193 − 0.193i)7-s − 5.40·9-s + (3.65 + 3.65i)11-s + (−0.193 − 3.60i)13-s + (−8.25 + 8.25i)15-s + 0.899i·17-s + (0.753 − 0.753i)19-s + (−0.560 + 0.560i)21-s − 1.89·23-s + 11.2i·25-s + 6.97i·27-s − 5.41·29-s + (3.24 − 3.24i)31-s + ⋯
L(s)  = 1  − 1.67i·3-s + (−1.27 − 1.27i)5-s + (−0.0730 − 0.0730i)7-s − 1.80·9-s + (1.10 + 1.10i)11-s + (−0.0536 − 0.998i)13-s + (−2.13 + 2.13i)15-s + 0.218i·17-s + (0.172 − 0.172i)19-s + (−0.122 + 0.122i)21-s − 0.394·23-s + 2.24i·25-s + 1.34i·27-s − 1.00·29-s + (0.583 − 0.583i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.971 - 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101645 + 0.841693i\)
\(L(\frac12)\) \(\approx\) \(0.101645 + 0.841693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.193 + 3.60i)T \)
good3 \( 1 + 2.89iT - 3T^{2} \)
5 \( 1 + (2.84 + 2.84i)T + 5iT^{2} \)
7 \( 1 + (0.193 + 0.193i)T + 7iT^{2} \)
11 \( 1 + (-3.65 - 3.65i)T + 11iT^{2} \)
17 \( 1 - 0.899iT - 17T^{2} \)
19 \( 1 + (-0.753 + 0.753i)T - 19iT^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 + 5.41T + 29T^{2} \)
31 \( 1 + (-3.24 + 3.24i)T - 31iT^{2} \)
37 \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \)
41 \( 1 + (5.79 + 5.79i)T + 41iT^{2} \)
43 \( 1 + 2.81T + 43T^{2} \)
47 \( 1 + (-7.49 - 7.49i)T + 47iT^{2} \)
53 \( 1 + 5.69T + 53T^{2} \)
59 \( 1 + (8.44 + 8.44i)T + 59iT^{2} \)
61 \( 1 + 2.98T + 61T^{2} \)
67 \( 1 + (-8.85 + 8.85i)T - 67iT^{2} \)
71 \( 1 + (1.91 - 1.91i)T - 71iT^{2} \)
73 \( 1 + (-5.40 + 5.40i)T - 73iT^{2} \)
79 \( 1 + 1.20iT - 79T^{2} \)
83 \( 1 + (-2.14 + 2.14i)T - 83iT^{2} \)
89 \( 1 + (1.79 - 1.79i)T - 89iT^{2} \)
97 \( 1 + (1.10 + 1.10i)T + 97iT^{2} \)
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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

−11.18862572772554699572457645746, −9.543394580411627421256449610802, −8.580789252972954256477759193705, −7.77647499530995378190382694960, −7.29707853932406437398705096522, −6.14309207174522921088001003400, −4.86787761203871120217544459505, −3.66705482655324841612240260755, −1.79983040144133385598145312036, −0.56237364290538519503156512407, 3.07994026322223811252007490085, 3.75704504484677243408496555543, 4.49956663509706544613392944053, 6.01930872444795930435241932225, 6.97204857223458824460476486090, 8.239871368134428199830429468701, 9.099274297169274335029615804080, 9.992287753099839875410767762931, 10.90106618460086750326296212694, 11.46037513570913868533151646382

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