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Properties

Label 2-3204-1068.383-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.729 + 0.683i$
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  + (−0.989 − 0.142i)2-s + (0.959 + 0.281i)4-s + (1.01 − 0.377i)5-s + (−0.909 − 0.415i)8-s + (−1.05 + 0.229i)10-s + (1.53 − 1.23i)13-s + (0.841 + 0.540i)16-s + (−0.0855 + 0.114i)17-s + (1.07 − 0.0771i)20-s + (0.127 − 0.110i)25-s + (−1.69 + 1.00i)26-s + (0.0225 − 0.631i)29-s + (−0.755 − 0.654i)32-s + (0.100 − 0.100i)34-s + (0.658 + 1.58i)37-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (0.959 + 0.281i)4-s + (1.01 − 0.377i)5-s + (−0.909 − 0.415i)8-s + (−1.05 + 0.229i)10-s + (1.53 − 1.23i)13-s + (0.841 + 0.540i)16-s + (−0.0855 + 0.114i)17-s + (1.07 − 0.0771i)20-s + (0.127 − 0.110i)25-s + (−1.69 + 1.00i)26-s + (0.0225 − 0.631i)29-s + (−0.755 − 0.654i)32-s + (0.100 − 0.100i)34-s + (0.658 + 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.729 + 0.683i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (2519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.729 + 0.683i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050932643\)
\(L(\frac12)\) \(\approx\) \(1.050932643\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 \)
89 \( 1 + (-0.349 + 0.936i)T \)
good5 \( 1 + (-1.01 + 0.377i)T + (0.755 - 0.654i)T^{2} \)
7 \( 1 + (-0.0713 + 0.997i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (-1.53 + 1.23i)T + (0.212 - 0.977i)T^{2} \)
17 \( 1 + (0.0855 - 0.114i)T + (-0.281 - 0.959i)T^{2} \)
19 \( 1 + (-0.349 - 0.936i)T^{2} \)
23 \( 1 + (0.936 - 0.349i)T^{2} \)
29 \( 1 + (-0.0225 + 0.631i)T + (-0.997 - 0.0713i)T^{2} \)
31 \( 1 + (-0.936 - 0.349i)T^{2} \)
37 \( 1 + (-0.658 - 1.58i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.21 + 0.977i)T + (0.212 + 0.977i)T^{2} \)
43 \( 1 + (-0.997 + 0.0713i)T^{2} \)
47 \( 1 + (0.540 - 0.841i)T^{2} \)
53 \( 1 + (-1.05 - 0.574i)T + (0.540 + 0.841i)T^{2} \)
59 \( 1 + (-0.977 + 0.212i)T^{2} \)
61 \( 1 + (0.655 + 1.30i)T + (-0.599 + 0.800i)T^{2} \)
67 \( 1 + (-0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.755 + 0.654i)T^{2} \)
73 \( 1 + (1.03 - 1.61i)T + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.909 - 0.415i)T^{2} \)
83 \( 1 + (0.877 + 0.479i)T^{2} \)
97 \( 1 + (-0.398 + 0.871i)T + (-0.654 - 0.755i)T^{2} \)
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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

−8.608454321524327191580449599659, −8.365577550877702054007552965678, −7.43152197803247768255074090244, −6.43239757148239040424352165356, −5.93731484330594010150968098025, −5.21107419476142471673892834764, −3.80719346631303904384580419145, −2.94771304086443363990614093318, −1.88647772606716394131275255716, −0.999524381125432262311132563490, 1.35498652951570051403332991689, 2.08813826925448337804775262179, 3.11108452508635094421733904537, 4.21356933380488879932809197844, 5.51268624354653468990745477074, 6.16302292553000124884854444479, 6.65456656151156006239192933717, 7.42672572613348324944419570423, 8.389027553330450522696278036811, 9.040044400703523627663812477947

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