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Properties

Label 2-416-104.29-c1-0-9
Degree $2$
Conductor $416$
Sign $-0.940 + 0.339i$
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  + (−1.47 + 0.852i)3-s − 2.59i·5-s + (0.300 − 0.520i)7-s + (−0.0479 + 0.0830i)9-s + (−2.40 + 1.39i)11-s + (−3.60 + 0.0531i)13-s + (2.21 + 3.83i)15-s + (−1.29 + 2.24i)17-s + (−4.38 − 2.52i)19-s + 1.02i·21-s + (−2.94 − 5.09i)23-s − 1.75·25-s − 5.27i·27-s + (1.54 − 0.889i)29-s − 5.09·31-s + ⋯
L(s)  = 1  + (−0.852 + 0.491i)3-s − 1.16i·5-s + (0.113 − 0.196i)7-s + (−0.0159 + 0.0276i)9-s + (−0.726 + 0.419i)11-s + (−0.999 + 0.0147i)13-s + (0.571 + 0.990i)15-s + (−0.313 + 0.543i)17-s + (−1.00 − 0.580i)19-s + 0.223i·21-s + (−0.613 − 1.06i)23-s − 0.351·25-s − 1.01i·27-s + (0.286 − 0.165i)29-s − 0.914·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.940 + 0.339i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0314200 - 0.179874i\)
\(L(\frac12)\) \(\approx\) \(0.0314200 - 0.179874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 - 0.0531i)T \)
good3 \( 1 + (1.47 - 0.852i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 2.59iT - 5T^{2} \)
7 \( 1 + (-0.300 + 0.520i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.40 - 1.39i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.29 - 2.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.38 + 2.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.94 + 5.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.54 + 0.889i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.09T + 31T^{2} \)
37 \( 1 + (8.82 - 5.09i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.26 + 9.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.44 - 4.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 + 6.01iT - 53T^{2} \)
59 \( 1 + (-9.44 - 5.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 - 1.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.39 - 1.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.230 - 0.399i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 - 7.83T + 79T^{2} \)
83 \( 1 - 0.930iT - 83T^{2} \)
89 \( 1 + (-1.17 - 2.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.91 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60233445720854464998791607134, −10.22025500459634980028112993749, −8.950281550232927859674414112422, −8.209164333444919191828077374895, −7.00726388058215682757275900562, −5.73027156925078693364598599709, −4.86804840888859154707591746289, −4.33380164459787532556390227673, −2.22542134888762906908384484182, −0.12011372704836230151846076526, 2.21447230111572449568480544150, 3.47814157870467479474050071289, 5.15328651748459233562633556187, 5.99744693731449219505200605886, 6.92861743034562093367102429841, 7.59676154872337684693754889138, 8.883906309947959837112781356518, 10.12806131364630895150025292241, 10.79755398585553615868115473821, 11.58583067018870928118558273811

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