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Properties

Label 2-3276-13.10-c1-0-14
Degree $2$
Conductor $3276$
Sign $0.996 + 0.0848i$
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  − 2.30i·5-s + (0.866 + 0.5i)7-s + (−1.21 + 0.701i)11-s + (−1.57 − 3.24i)13-s + (−2.96 + 5.13i)17-s + (5.10 + 2.94i)19-s + (2.00 + 3.46i)23-s − 0.291·25-s + (0.712 + 1.23i)29-s + 6.67i·31-s + (1.15 − 1.99i)35-s + (2.42 − 1.39i)37-s + (2.47 − 1.42i)41-s + (5.47 − 9.47i)43-s + 12.5i·47-s + ⋯
L(s)  = 1  − 1.02i·5-s + (0.327 + 0.188i)7-s + (−0.366 + 0.211i)11-s + (−0.436 − 0.899i)13-s + (−0.719 + 1.24i)17-s + (1.17 + 0.676i)19-s + (0.417 + 0.722i)23-s − 0.0582·25-s + (0.132 + 0.229i)29-s + 1.19i·31-s + (0.194 − 0.336i)35-s + (0.397 − 0.229i)37-s + (0.386 − 0.223i)41-s + (0.834 − 1.44i)43-s + 1.83i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.996 + 0.0848i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.996 + 0.0848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.832868460\)
\(L(\frac12)\) \(\approx\) \(1.832868460\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 3.24i)T \)
good5 \( 1 + 2.30iT - 5T^{2} \)
11 \( 1 + (1.21 - 0.701i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.96 - 5.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.10 - 2.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.00 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.712 - 1.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.67iT - 31T^{2} \)
37 \( 1 + (-2.42 + 1.39i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.47 + 1.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.47 + 9.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 - 5.14T + 53T^{2} \)
59 \( 1 + (-9.08 - 5.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.72 + 9.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.18 + 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.9 + 6.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.30iT - 73T^{2} \)
79 \( 1 - 2.50T + 79T^{2} \)
83 \( 1 + 18.0iT - 83T^{2} \)
89 \( 1 + (-8.44 + 4.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.06 + 1.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.797603286815609609026272291194, −7.81436039817637731336640915319, −7.44855625798252921466434489509, −6.22424198975215366917148868330, −5.40106800518176570185208073614, −4.98431039991796905687123243185, −4.02338156874057827066937508905, −3.05167526703688199297854271429, −1.87856190907627492600453289993, −0.899835538958450262017491475564, 0.74870298641834132247384944525, 2.43092162030006610192790656090, 2.75970734906246451940207162600, 4.01224473249004059059234639091, 4.78724408887680170172954923229, 5.58987036227110575221454142596, 6.73599617935228280193844311310, 7.01295545355905263967317322292, 7.73979447922337763747868916247, 8.685068986792229289532746917798

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