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Properties

Label 2-3276-13.4-c1-0-3
Degree $2$
Conductor $3276$
Sign $-0.496 - 0.867i$
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  + 0.968i·5-s + (0.866 − 0.5i)7-s + (−2.27 − 1.31i)11-s + (−3.60 − 0.0604i)13-s + (−0.986 − 1.70i)17-s + (3.33 − 1.92i)19-s + (−1.68 + 2.91i)23-s + 4.06·25-s + (−2.56 + 4.44i)29-s + 2.80i·31-s + (0.484 + 0.838i)35-s + (−0.265 − 0.153i)37-s + (−2.68 − 1.55i)41-s + (1.78 + 3.09i)43-s + 11.8i·47-s + ⋯
L(s)  = 1  + 0.433i·5-s + (0.327 − 0.188i)7-s + (−0.685 − 0.395i)11-s + (−0.999 − 0.0167i)13-s + (−0.239 − 0.414i)17-s + (0.764 − 0.441i)19-s + (−0.351 + 0.608i)23-s + 0.812·25-s + (−0.476 + 0.826i)29-s + 0.503i·31-s + (0.0818 + 0.141i)35-s + (−0.0436 − 0.0251i)37-s + (−0.419 − 0.242i)41-s + (0.272 + 0.472i)43-s + 1.72i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.496 - 0.867i$
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: \((2,\ 3276,\ (\ :1/2),\ -0.496 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9029381572\)
\(L(\frac12)\) \(\approx\) \(0.9029381572\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.60 + 0.0604i)T \)
good5 \( 1 - 0.968iT - 5T^{2} \)
11 \( 1 + (2.27 + 1.31i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.986 + 1.70i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.33 + 1.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.68 - 2.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.56 - 4.44i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.80iT - 31T^{2} \)
37 \( 1 + (0.265 + 0.153i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.68 + 1.55i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.78 - 3.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.8iT - 47T^{2} \)
53 \( 1 + 4.44T + 53T^{2} \)
59 \( 1 + (1.47 - 0.853i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.20 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.09 - 1.20i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (13.6 - 7.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.47iT - 73T^{2} \)
79 \( 1 - 0.738T + 79T^{2} \)
83 \( 1 - 5.59iT - 83T^{2} \)
89 \( 1 + (6.77 + 3.91i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.75 + 4.47i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.911146766727701772617742377690, −8.047295397157054024237466282273, −7.29773140182071923440506635860, −6.92081485205352061224416752575, −5.72678533252942311919463497623, −5.12974124344878887824108652063, −4.33626957759550742204033710017, −3.13173010699035818333028082874, −2.59852409253865668274120300153, −1.24944990725363801973922542617, 0.27699417621681705043920812998, 1.78502187748449291308413981852, 2.57508211382709089248288594086, 3.71782704132540311574855757001, 4.72845347726372881724969148649, 5.17635481057688505659181646708, 6.06433434739409060171416952430, 7.00926102238274508368347591543, 7.75418825779812731018261405908, 8.293582660036717933512047835679

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