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Properties

Label 2-416-104.11-c1-0-10
Degree $2$
Conductor $416$
Sign $-0.849 + 0.527i$
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  + (−0.0597 − 0.103i)3-s + (−2.08 + 2.08i)5-s + (−1.83 + 0.491i)7-s + (1.49 − 2.58i)9-s + (−5.53 − 1.48i)11-s + (0.0282 − 3.60i)13-s + (0.340 + 0.0913i)15-s + (−3.70 − 2.13i)17-s + (4.12 − 1.10i)19-s + (0.160 + 0.160i)21-s + (−1.56 − 2.70i)23-s − 3.72i·25-s − 0.715·27-s + (−3.41 + 1.97i)29-s + (−5.91 + 5.91i)31-s + ⋯
L(s)  = 1  + (−0.0344 − 0.0597i)3-s + (−0.934 + 0.934i)5-s + (−0.693 + 0.185i)7-s + (0.497 − 0.861i)9-s + (−1.66 − 0.446i)11-s + (0.00782 − 0.999i)13-s + (0.0880 + 0.0235i)15-s + (−0.897 − 0.518i)17-s + (0.946 − 0.253i)19-s + (0.0350 + 0.0350i)21-s + (−0.325 − 0.563i)23-s − 0.745i·25-s − 0.137·27-s + (−0.634 + 0.366i)29-s + (−1.06 + 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.849 + 0.527i$
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: \((2,\ 416,\ (\ :1/2),\ -0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0610533 - 0.213996i\)
\(L(\frac12)\) \(\approx\) \(0.0610533 - 0.213996i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0282 + 3.60i)T \)
good3 \( 1 + (0.0597 + 0.103i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.08 - 2.08i)T - 5iT^{2} \)
7 \( 1 + (1.83 - 0.491i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.53 + 1.48i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.70 + 2.13i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.12 + 1.10i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.56 + 2.70i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.41 - 1.97i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.91 - 5.91i)T - 31iT^{2} \)
37 \( 1 + (0.0218 - 0.0814i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.89 - 7.07i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.96 - 2.28i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.33 - 1.33i)T + 47iT^{2} \)
53 \( 1 + 7.65iT - 53T^{2} \)
59 \( 1 + (0.332 + 1.23i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.12 + 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.943 - 3.52i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.87 + 6.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.35 - 2.35i)T - 73iT^{2} \)
79 \( 1 + 4.48iT - 79T^{2} \)
83 \( 1 + (0.871 + 0.871i)T + 83iT^{2} \)
89 \( 1 + (0.761 + 0.204i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.8 + 3.18i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.83136610905134386353918849243, −10.11829826503382618448858807804, −9.043542094314991423204339600828, −7.83401023351452000154507038042, −7.20951971853189919703015286500, −6.22099859675046300523166166618, −5.01587347882311075681856569336, −3.45170685483473557780746556358, −2.87994524550062699555559670175, −0.13530578043297229424211231585, 2.09967550700940040725919181612, 3.84973400691633183750223291300, 4.68238085309089028128228035497, 5.68465384852507084837009857144, 7.31788577128033686607840914799, 7.72109871794912207651550986995, 8.839591858113367345709770906727, 9.783292211111754404344773578959, 10.69036758427617993263424511740, 11.58884375772164680767089512166

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