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Properties

Label 2-3204-1068.419-c0-0-1
Degree $2$
Conductor $3204$
Sign $-0.587 - 0.809i$
Analytic cond. $1.59900$
Root an. cond. $1.26451$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.841 − 0.540i)4-s + (0.129 − 1.81i)5-s + (−0.755 + 0.654i)8-s + (−1.70 − 0.635i)10-s + (−0.834 + 1.20i)13-s + (0.415 + 0.909i)16-s + (−1.40 − 0.767i)17-s + (−1.09 + 1.45i)20-s + (−2.28 − 0.328i)25-s + (0.918 + 1.13i)26-s + (−0.912 − 0.456i)29-s + (0.989 − 0.142i)32-s + (−1.13 + 1.13i)34-s + (−0.480 − 1.15i)37-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.841 − 0.540i)4-s + (0.129 − 1.81i)5-s + (−0.755 + 0.654i)8-s + (−1.70 − 0.635i)10-s + (−0.834 + 1.20i)13-s + (0.415 + 0.909i)16-s + (−1.40 − 0.767i)17-s + (−1.09 + 1.45i)20-s + (−2.28 − 0.328i)25-s + (0.918 + 1.13i)26-s + (−0.912 − 0.456i)29-s + (0.989 − 0.142i)32-s + (−1.13 + 1.13i)34-s + (−0.480 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $-0.587 - 0.809i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (2555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ -0.587 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6896409271\)
\(L(\frac12)\) \(\approx\) \(0.6896409271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 + 0.959i)T \)
3 \( 1 \)
89 \( 1 + (-0.997 + 0.0713i)T \)
good5 \( 1 + (-0.129 + 1.81i)T + (-0.989 - 0.142i)T^{2} \)
7 \( 1 + (0.800 - 0.599i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (0.834 - 1.20i)T + (-0.349 - 0.936i)T^{2} \)
17 \( 1 + (1.40 + 0.767i)T + (0.540 + 0.841i)T^{2} \)
19 \( 1 + (-0.997 - 0.0713i)T^{2} \)
23 \( 1 + (0.0713 - 0.997i)T^{2} \)
29 \( 1 + (0.912 + 0.456i)T + (0.599 + 0.800i)T^{2} \)
31 \( 1 + (-0.0713 - 0.997i)T^{2} \)
37 \( 1 + (0.480 + 1.15i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.650 + 0.936i)T + (-0.349 + 0.936i)T^{2} \)
43 \( 1 + (0.599 - 0.800i)T^{2} \)
47 \( 1 + (0.909 - 0.415i)T^{2} \)
53 \( 1 + (-1.71 - 0.373i)T + (0.909 + 0.415i)T^{2} \)
59 \( 1 + (-0.936 - 0.349i)T^{2} \)
61 \( 1 + (0.611 + 0.156i)T + (0.877 + 0.479i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.989 + 0.142i)T^{2} \)
73 \( 1 + (-1.53 + 0.698i)T + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (0.755 + 0.654i)T^{2} \)
83 \( 1 + (-0.977 - 0.212i)T^{2} \)
97 \( 1 + (-0.278 - 0.321i)T + (-0.142 + 0.989i)T^{2} \)
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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

−8.849817997677112931607079543653, −7.83440947666245813278216649184, −6.80788208456516361428451858168, −5.70633162942980657815910822742, −5.01706886411076839718409000059, −4.46883283393106670866656732779, −3.85143717969677778815849046889, −2.31786320741198611640679284955, −1.74850398787531372274972183171, −0.35459057993857070368888142227, 2.25403950844940337284661544714, 3.13519161703217269899774248017, 3.79754156636927992258651305856, 4.91577054204361984163322998927, 5.71003239349443295588580246693, 6.58837400802465096669523903352, 6.85076131601468152980701088157, 7.71098128783134700229167069995, 8.265857250132607772993146751511, 9.261367312934358856303284147640

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