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Properties

Label 2-3204-1068.407-c0-0-0
Degree $2$
Conductor $3204$
Sign $0.165 + 0.986i$
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  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (0.550 + 0.119i)5-s + (−0.540 − 0.841i)8-s + (0.494 − 0.270i)10-s + (0.183 − 0.109i)13-s + (−0.959 − 0.281i)16-s + (1.86 − 0.133i)17-s + (0.196 − 0.527i)20-s + (−0.620 − 0.283i)25-s + (0.0675 − 0.202i)26-s + (1.11 + 0.777i)29-s + (−0.909 + 0.415i)32-s + (1.32 − 1.32i)34-s + (−0.583 + 0.241i)37-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (0.550 + 0.119i)5-s + (−0.540 − 0.841i)8-s + (0.494 − 0.270i)10-s + (0.183 − 0.109i)13-s + (−0.959 − 0.281i)16-s + (1.86 − 0.133i)17-s + (0.196 − 0.527i)20-s + (−0.620 − 0.283i)25-s + (0.0675 − 0.202i)26-s + (1.11 + 0.777i)29-s + (−0.909 + 0.415i)32-s + (1.32 − 1.32i)34-s + (−0.583 + 0.241i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (1475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.141159643\)
\(L(\frac12)\) \(\approx\) \(2.141159643\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.755 + 0.654i)T \)
3 \( 1 \)
89 \( 1 + (-0.212 - 0.977i)T \)
good5 \( 1 + (-0.550 - 0.119i)T + (0.909 + 0.415i)T^{2} \)
7 \( 1 + (0.936 - 0.349i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.183 + 0.109i)T + (0.479 - 0.877i)T^{2} \)
17 \( 1 + (-1.86 + 0.133i)T + (0.989 - 0.142i)T^{2} \)
19 \( 1 + (-0.212 + 0.977i)T^{2} \)
23 \( 1 + (-0.977 - 0.212i)T^{2} \)
29 \( 1 + (-1.11 - 0.777i)T + (0.349 + 0.936i)T^{2} \)
31 \( 1 + (0.977 - 0.212i)T^{2} \)
37 \( 1 + (0.583 - 0.241i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.47 + 0.877i)T + (0.479 + 0.877i)T^{2} \)
43 \( 1 + (0.349 - 0.936i)T^{2} \)
47 \( 1 + (-0.281 + 0.959i)T^{2} \)
53 \( 1 + (-1.19 + 1.59i)T + (-0.281 - 0.959i)T^{2} \)
59 \( 1 + (-0.877 + 0.479i)T^{2} \)
61 \( 1 + (0.0126 + 0.354i)T + (-0.997 + 0.0713i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.909 - 0.415i)T^{2} \)
73 \( 1 + (-0.0801 + 0.273i)T + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.540 - 0.841i)T^{2} \)
83 \( 1 + (0.599 - 0.800i)T^{2} \)
97 \( 1 + (1.34 - 0.865i)T + (0.415 - 0.909i)T^{2} \)
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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

−8.803830568020912493717973603331, −7.954556158000743585696911018006, −6.93010819629685520492254637004, −6.27349302469488356248147204445, −5.40513055925232430460228515429, −5.00251805946206440096971288132, −3.74911643861923558174343100049, −3.19035820040627350771209167603, −2.14301208225152337360290548815, −1.15746987399119228122309674354, 1.55670670046771357078653226321, 2.80055285425460696869023113605, 3.56762363806097269611569354962, 4.49073986428972093168753009786, 5.38130949067644478335627821049, 5.86748277281505812121086521916, 6.62826946521136586778637374012, 7.47178033504516117020392849260, 8.122866004474904979895095891033, 8.785808250287553235201786020327

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