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Properties

Label 2-3276-13.10-c1-0-12
Degree $2$
Conductor $3276$
Sign $-0.446 - 0.894i$
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  + 3.22i·5-s + (0.866 + 0.5i)7-s + (−1.17 + 0.678i)11-s + (1.95 − 3.03i)13-s + (−1.44 + 2.50i)17-s + (5.05 + 2.92i)19-s + (0.482 + 0.835i)23-s − 5.39·25-s + (3.61 + 6.26i)29-s − 2.60i·31-s + (−1.61 + 2.79i)35-s + (−4.35 + 2.51i)37-s + (8.03 − 4.63i)41-s + (0.226 − 0.392i)43-s + 5.42i·47-s + ⋯
L(s)  = 1  + 1.44i·5-s + (0.327 + 0.188i)7-s + (−0.354 + 0.204i)11-s + (0.541 − 0.840i)13-s + (−0.351 + 0.608i)17-s + (1.16 + 0.669i)19-s + (0.100 + 0.174i)23-s − 1.07·25-s + (0.672 + 1.16i)29-s − 0.468i·31-s + (−0.272 + 0.471i)35-s + (−0.715 + 0.413i)37-s + (1.25 − 0.724i)41-s + (0.0345 − 0.0597i)43-s + 0.791i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.446 - 0.894i$
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: \((2,\ 3276,\ (\ :1/2),\ -0.446 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.736743986\)
\(L(\frac12)\) \(\approx\) \(1.736743986\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 + 3.03i)T \)
good5 \( 1 - 3.22iT - 5T^{2} \)
11 \( 1 + (1.17 - 0.678i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.44 - 2.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.05 - 2.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.482 - 0.835i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.61 - 6.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.60iT - 31T^{2} \)
37 \( 1 + (4.35 - 2.51i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.03 + 4.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.226 + 0.392i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.42iT - 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 + (2.69 + 1.55i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.31 - 4.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.232 + 0.134i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.6 - 6.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.24iT - 73T^{2} \)
79 \( 1 + 8.13T + 79T^{2} \)
83 \( 1 - 2.59iT - 83T^{2} \)
89 \( 1 + (5.84 - 3.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.5 + 6.67i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.786900208582508526803347153905, −8.007283441063841712041877182777, −7.40456663185525336326357133723, −6.69996396994302279398049709297, −5.87245814861654236388179837074, −5.26970089693466565659212351543, −4.05003670646927070630669932925, −3.21422364286908981294099760357, −2.58458015539271588440802076633, −1.35191363878764940238097246791, 0.57025402115164352141845504660, 1.48018979723630319712962674001, 2.65840288494546494026273136037, 3.86678737375396869718616462893, 4.70641949597410112116939737256, 5.11671415657919898394414771198, 6.05867835262213709934843414493, 6.96246519023806239196627076703, 7.81854606601088230421911667684, 8.438632177714979027689648394236

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