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Properties

Label 2-3204-1068.647-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.484 + 0.874i$
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 + (1.11 − 0.777i)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 + (1.06 + 0.855i)26-s + (−0.769 − 1.53i)29-s + (0.989 + 0.142i)32-s + (−1.13 − 1.13i)34-s + (−1.43 − 0.595i)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 + (1.11 − 0.777i)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 + (1.06 + 0.855i)26-s + (−0.769 − 1.53i)29-s + (0.989 + 0.142i)32-s + (−1.13 − 1.13i)34-s + (−1.43 − 0.595i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.484 + 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9432902014\)
\(L(\frac12)\) \(\approx\) \(0.9432902014\)
\(L(1)\) 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 \( 1 + (-1.11 + 0.777i)T + (0.349 - 0.936i)T^{2} \)
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 \( 1 + (0.769 + 1.53i)T + (-0.599 + 0.800i)T^{2} \)
31 \( 1 + (0.0713 - 0.997i)T^{2} \)
37 \( 1 + (1.43 + 0.595i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.34 + 0.936i)T + (0.349 + 0.936i)T^{2} \)
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 \( 1 + (0.469 + 1.83i)T + (-0.877 + 0.479i)T^{2} \)
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

−8.517708121652241332923929691106, −8.236733326916244570176177076060, −7.28273859014833422612516784490, −6.29213677216928272265510400232, −5.61595216789704598504840681455, −5.05239596805760932516436195539, −4.10854661387306749459524507963, −3.71096742905495637508432908189, −1.92737081469762801932865803649, −0.51466032379147674073639213983, 1.69658927813540768881033688251, 2.56277035000424391414405972906, 3.39052697080287044095114455694, 3.94984065273511554142558598922, 4.99478261827239660320252708472, 6.00544402414599618430990917747, 6.77255033048405178011670074384, 7.18785252873815808105325983183, 8.552238945317020742692701351677, 8.990209856729822865330859311170

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