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Properties

Label 2-3276-13.4-c1-0-6
Degree $2$
Conductor $3276$
Sign $-0.378 - 0.925i$
Analytic cond. $26.1589$
Root an. cond. $5.11458$
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  + 1.46i·5-s + (−0.866 + 0.5i)7-s + (−2.38 − 1.37i)11-s + (3.56 + 0.531i)13-s + (−2.39 − 4.15i)17-s + (−1.70 + 0.985i)19-s + (1.46 − 2.54i)23-s + 2.85·25-s + (−2.73 + 4.73i)29-s + 3.90i·31-s + (−0.733 − 1.26i)35-s + (1.87 + 1.08i)37-s + (8.18 + 4.72i)41-s + (5.80 + 10.0i)43-s + 10.3i·47-s + ⋯
L(s)  = 1  + 0.655i·5-s + (−0.327 + 0.188i)7-s + (−0.718 − 0.414i)11-s + (0.989 + 0.147i)13-s + (−0.582 − 1.00i)17-s + (−0.391 + 0.226i)19-s + (0.306 − 0.530i)23-s + 0.570·25-s + (−0.507 + 0.879i)29-s + 0.700i·31-s + (−0.123 − 0.214i)35-s + (0.308 + 0.178i)37-s + (1.27 + 0.737i)41-s + (0.885 + 1.53i)43-s + 1.51i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(26.1589\)
Root analytic conductor: \(5.11458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3276} (2773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3276,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.161438428\)
\(L(\frac12)\) \(\approx\) \(1.161438428\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.56 - 0.531i)T \)
good5 \( 1 - 1.46iT - 5T^{2} \)
11 \( 1 + (2.38 + 1.37i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.39 + 4.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.70 - 0.985i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.46 + 2.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.73 - 4.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.90iT - 31T^{2} \)
37 \( 1 + (-1.87 - 1.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.18 - 4.72i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.80 - 10.0i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 1.24T + 53T^{2} \)
59 \( 1 + (3.95 - 2.28i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.19 + 7.27i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.25 + 1.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.58 - 4.95i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.93iT - 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 0.811iT - 83T^{2} \)
89 \( 1 + (-8.38 - 4.83i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.23 - 4.75i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.995350266734057407002171131420, −8.085809484800860141392277661104, −7.39129684463666566995775082474, −6.49413872011367769765292562187, −6.08813398017682716179804700863, −5.04713928489105187420946033434, −4.23481704709797503130528483120, −3.03731812278832047250876919517, −2.71215053159404284731625829394, −1.20332869589576936635470190648, 0.37625078260148654967251345728, 1.66935938057994066818196207646, 2.68313450375786750735177026297, 3.90296207928110072006215359191, 4.36808803087003196684841962014, 5.52447426511927613764613010924, 5.97522350671891982157732689048, 6.99650270430023907907683882437, 7.68300201746910111997818293582, 8.559642346249128106735540552189

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