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Properties

Label 2-416-13.9-c1-0-8
Degree $2$
Conductor $416$
Sign $0.0128 + 0.999i$
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  + (−1.65 + 2.87i)3-s + (−1.65 − 2.87i)7-s + (−4 − 6.92i)9-s + (−1.65 + 2.87i)11-s + (−1 − 3.46i)13-s + (1.5 + 2.59i)17-s + (−1.65 − 2.87i)19-s + 11·21-s + (1.65 − 2.87i)23-s − 5·25-s + 16.5·27-s + (2.5 − 4.33i)29-s + (−5.5 − 9.52i)33-s + (−4.5 + 7.79i)37-s + (11.6 + 2.87i)39-s + ⋯
L(s)  = 1  + (−0.957 + 1.65i)3-s + (−0.626 − 1.08i)7-s + (−1.33 − 2.30i)9-s + (−0.500 + 0.866i)11-s + (−0.277 − 0.960i)13-s + (0.363 + 0.630i)17-s + (−0.380 − 0.658i)19-s + 2.40·21-s + (0.345 − 0.598i)23-s − 25-s + 3.19·27-s + (0.464 − 0.804i)29-s + (−0.957 − 1.65i)33-s + (−0.739 + 1.28i)37-s + (1.85 + 0.459i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172266 - 0.170071i\)
\(L(\frac12)\) \(\approx\) \(0.172266 - 0.170071i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good3 \( 1 + (1.65 - 2.87i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (1.65 + 2.87i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.65 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.65 + 2.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.65 + 2.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.97 + 8.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.63T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + (1.65 + 2.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.97 - 8.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.97 + 8.61i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 6.63T + 79T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.59228494195776539483679486489, −10.20942033477625921288845056072, −9.715342692044473904762367653723, −8.408443863253367011453782325409, −7.05266527011519654170626685934, −6.03063996788653119022733098805, −4.98119725872838407342100120594, −4.20983021190472638747178398866, −3.18986675445017056955313795116, −0.16880055800898607543769502140, 1.72082436000843088939663263624, 2.94841899390525123404647264971, 5.14897833310567816177935343407, 5.92359059338008495314646692299, 6.59656335498201544698006380011, 7.60017885510005422526492770268, 8.463771853109622890807897440744, 9.568903761745710862371818343130, 10.94805996058471822880669610983, 11.64841941371666837721278998082

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