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Properties

Label 2-3276-13.10-c1-0-20
Degree $2$
Conductor $3276$
Sign $0.520 + 0.853i$
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  + 0.484i·5-s + (−0.866 − 0.5i)7-s + (−4.83 + 2.79i)11-s + (−1.68 + 3.18i)13-s + (−0.943 + 1.63i)17-s + (−5.00 − 2.88i)19-s + (−4.09 − 7.10i)23-s + 4.76·25-s + (3.40 + 5.89i)29-s + 0.394i·31-s + (0.242 − 0.419i)35-s + (9.16 − 5.29i)37-s + (9.31 − 5.37i)41-s + (4.55 − 7.89i)43-s − 7.20i·47-s + ⋯
L(s)  = 1  + 0.216i·5-s + (−0.327 − 0.188i)7-s + (−1.45 + 0.841i)11-s + (−0.468 + 0.883i)13-s + (−0.228 + 0.396i)17-s + (−1.14 − 0.662i)19-s + (−0.854 − 1.48i)23-s + 0.952·25-s + (0.632 + 1.09i)29-s + 0.0708i·31-s + (0.0409 − 0.0709i)35-s + (1.50 − 0.869i)37-s + (1.45 − 0.839i)41-s + (0.695 − 1.20i)43-s − 1.05i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.520 + 0.853i$
Analytic conductor: \(26.1589\)
Root analytic conductor: \(5.11458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3276} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3276,\ (\ :1/2),\ 0.520 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9977228084\)
\(L(\frac12)\) \(\approx\) \(0.9977228084\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 3.18i)T \)
good5 \( 1 - 0.484iT - 5T^{2} \)
11 \( 1 + (4.83 - 2.79i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.943 - 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.00 + 2.88i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 + 7.10i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.40 - 5.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.394iT - 31T^{2} \)
37 \( 1 + (-9.16 + 5.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.31 + 5.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.55 + 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.20iT - 47T^{2} \)
53 \( 1 + 4.98T + 53T^{2} \)
59 \( 1 + (-6.87 - 3.97i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.76 + 9.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.24 + 0.718i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.44 - 2.56i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.370iT - 73T^{2} \)
79 \( 1 - 4.36T + 79T^{2} \)
83 \( 1 + 5.00iT - 83T^{2} \)
89 \( 1 + (11.5 - 6.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.45 + 2.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.552610986142103437136821438225, −7.72952590940846055632267847257, −6.92472660444704263538887435607, −6.50862001984814895325235834306, −5.40440091173816318090576173379, −4.58322788342473552155178479408, −4.00584272438689226185529458444, −2.51525633453300907289036332937, −2.27066274850468854423095103552, −0.37486064133751760326021913956, 0.881268490792898029937102007509, 2.50509279631856693948458033042, 2.94874274202682093335991006310, 4.16167414455108618816846489934, 4.98524829066129873066426424946, 5.86901836863556803755250390400, 6.22845106167722240978859643488, 7.58822999044682579940039250022, 7.965374184237043622729590328099, 8.577997671873982064741734288978

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