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Properties

Label 2-416-13.9-c1-0-5
Degree $2$
Conductor $416$
Sign $0.993 - 0.116i$
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  + (0.599 − 1.03i)3-s + 1.56·5-s + (1.53 + 2.66i)7-s + (0.780 + 1.35i)9-s + (−2.73 + 4.73i)11-s + (3.34 − 1.35i)13-s + (0.936 − 1.62i)15-s + (1.06 + 1.83i)17-s + (−3.67 − 6.35i)19-s + 3.68·21-s + (1.79 − 3.11i)23-s − 2.56·25-s + 5.47·27-s + (2.5 − 4.33i)29-s − 6.67·31-s + ⋯
L(s)  = 1  + (0.346 − 0.599i)3-s + 0.698·5-s + (0.580 + 1.00i)7-s + (0.260 + 0.450i)9-s + (−0.824 + 1.42i)11-s + (0.926 − 0.375i)13-s + (0.241 − 0.418i)15-s + (0.257 + 0.445i)17-s + (−0.842 − 1.45i)19-s + 0.804·21-s + (0.375 − 0.649i)23-s − 0.512·25-s + 1.05·27-s + (0.464 − 0.804i)29-s − 1.19·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.993 - 0.116i$
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: \((2,\ 416,\ (\ :1/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80336 + 0.104951i\)
\(L(\frac12)\) \(\approx\) \(1.80336 + 0.104951i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.34 + 1.35i)T \)
good3 \( 1 + (-0.599 + 1.03i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.56T + 5T^{2} \)
7 \( 1 + (-1.53 - 2.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.06 - 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.67 + 6.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.79 + 3.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + (-3.06 + 5.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.06 - 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.336 - 0.583i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.525T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 + (5.13 + 8.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.62 + 4.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.47 - 4.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.67 - 6.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16.6T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 - 8.54T + 83T^{2} \)
89 \( 1 + (-0.842 + 1.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.40 + 9.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.13005317468900341295673218474, −10.37159971198266096263764138229, −9.313721966537304198844016705439, −8.406585032564876468205785615369, −7.64307195770352831188042693498, −6.55747016909679500672543117618, −5.46490284142844472566588320409, −4.55165130598770223806425384934, −2.51573433466867128932413780597, −1.88801661508749921013684767836, 1.37853139296822822563482489152, 3.27440240414119681047159960488, 4.10461914382757405090503890863, 5.44313348125259165748857446268, 6.34559881117562148553591268039, 7.64666424807685433172425869992, 8.553128955743241909124126323729, 9.393880952901912407239673180502, 10.53696606507583770699863216908, 10.72833156119298176571505671884

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