↑

Properties

Label 2-3276-13.10-c1-0-25
Degree $2$
Conductor $3276$
Sign $0.283 + 0.958i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.91i·5-s + (0.866 + 0.5i)7-s + (−2.71 + 1.56i)11-s + (2.44 − 2.64i)13-s + (−1.02 + 1.78i)17-s + (−4.35 − 2.51i)19-s + (−3.46 − 6.00i)23-s + 1.32·25-s + (−5.03 − 8.71i)29-s − 10.5i·31-s + (−0.959 + 1.66i)35-s + (−0.508 + 0.293i)37-s + (−1.22 + 0.709i)41-s + (−1.39 + 2.41i)43-s + 3.70i·47-s + ⋯
L(s)  = 1  + 0.857i·5-s + (0.327 + 0.188i)7-s + (−0.819 + 0.472i)11-s + (0.679 − 0.734i)13-s + (−0.249 + 0.431i)17-s + (−0.998 − 0.576i)19-s + (−0.723 − 1.25i)23-s + 0.264·25-s + (−0.934 − 1.61i)29-s − 1.89i·31-s + (−0.162 + 0.280i)35-s + (−0.0835 + 0.0482i)37-s + (−0.191 + 0.110i)41-s + (−0.212 + 0.367i)43-s + 0.540i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3276\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.283 + 0.958i$
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: \((2,\ 3276,\ (\ :1/2),\ 0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.148745395\)
\(L(\frac12)\) \(\approx\) \(1.148745395\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.44 + 2.64i)T \)
good5 \( 1 - 1.91iT - 5T^{2} \)
11 \( 1 + (2.71 - 1.56i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.02 - 1.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.35 + 2.51i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.46 + 6.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.03 + 8.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5iT - 31T^{2} \)
37 \( 1 + (0.508 - 0.293i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.22 - 0.709i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.39 - 2.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.70iT - 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + (-3.93 - 2.27i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.20 + 2.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.7 + 7.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.32 + 3.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.18iT - 73T^{2} \)
79 \( 1 - 0.00212T + 79T^{2} \)
83 \( 1 - 9.23iT - 83T^{2} \)
89 \( 1 + (-12.3 + 7.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.08 + 0.625i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.183274697643044424567408404588, −7.979067014090404948898955743454, −6.94994163148071810106375312140, −6.22216591172019134133341625190, −5.62153396847248070368631833735, −4.50866926966198205622740229679, −3.85181506598952364713662971237, −2.58556396090839606693180466999, −2.18122558740481675773803010492, −0.36077562533224782490899938757, 1.18461830890709477909926426467, 2.05052667464847936685450321158, 3.41026565831203392200005306598, 4.10284016618305827074012969772, 5.18511717955320488445138010130, 5.46386749899628048719733068646, 6.64039313024853596226808726781, 7.29168061336178818757894075684, 8.311843036763357139552918770456, 8.668024742083726655446218611703

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