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Properties

Label 2-3276-13.4-c1-0-22
Degree $2$
Conductor $3276$
Sign $0.550 + 0.834i$
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  − 2.21i·5-s + (0.866 − 0.5i)7-s + (−0.0357 − 0.0206i)11-s + (3.60 − 0.167i)13-s + (1.18 + 2.05i)17-s + (3.42 − 1.97i)19-s + (3.23 − 5.60i)23-s + 0.0938·25-s + (−2.70 + 4.68i)29-s + 5.19i·31-s + (−1.10 − 1.91i)35-s + (9.80 + 5.66i)37-s + (−9.94 − 5.74i)41-s + (−0.735 − 1.27i)43-s + 2.95i·47-s + ⋯
L(s)  = 1  − 0.990i·5-s + (0.327 − 0.188i)7-s + (−0.0107 − 0.00622i)11-s + (0.998 − 0.0463i)13-s + (0.288 + 0.499i)17-s + (0.786 − 0.454i)19-s + (0.675 − 1.16i)23-s + 0.0187·25-s + (−0.502 + 0.869i)29-s + 0.932i·31-s + (−0.187 − 0.324i)35-s + (1.61 + 0.930i)37-s + (−1.55 − 0.896i)41-s + (−0.112 − 0.194i)43-s + 0.431i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.550 + 0.834i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.550 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.147444066\)
\(L(\frac12)\) \(\approx\) \(2.147444066\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.167i)T \)
good5 \( 1 + 2.21iT - 5T^{2} \)
11 \( 1 + (0.0357 + 0.0206i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.18 - 2.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.42 + 1.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.23 + 5.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.70 - 4.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + (-9.80 - 5.66i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.94 + 5.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.735 + 1.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.95iT - 47T^{2} \)
53 \( 1 + 2.27T + 53T^{2} \)
59 \( 1 + (-6.55 + 3.78i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.386 + 0.670i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.33 + 0.772i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.51 + 2.02i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.00iT - 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (-4.42 - 2.55i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.10 - 2.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.551956784722341469343807358310, −8.001951578852940976323327214403, −6.99596426240471645672973909791, −6.30498960236011166761706737273, −5.24216942483238150878074966787, −4.85754831190039602016787104283, −3.86102478882927466062480865094, −2.99020438975252012282706342289, −1.59476554263807352974093589387, −0.821861966547071599481817855983, 1.09887257209919745387393044564, 2.29775007003915917429523166223, 3.23631519431669077440864624999, 3.88216114284357997567319135831, 5.04245536226097847368549514305, 5.82859180998472072156610592222, 6.46323817150372698280225693158, 7.41900055907727293920333776246, 7.80725020698387511493150581439, 8.766055210218465655180726392696

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