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Properties

Label 2-3204-1068.431-c0-0-1
Degree $2$
Conductor $3204$
Sign $-0.638 + 0.769i$
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  + (−0.909 − 0.415i)2-s + (0.654 + 0.755i)4-s + (0.948 − 1.73i)5-s + (−0.281 − 0.959i)8-s + (−1.58 + 1.18i)10-s + (−0.912 + 0.456i)13-s + (−0.142 + 0.989i)16-s + (−0.398 − 0.148i)17-s + (1.93 − 0.420i)20-s + (−1.57 − 2.45i)25-s + (1.01 − 0.0364i)26-s + (1.13 + 0.121i)29-s + (0.540 − 0.841i)32-s + (0.300 + 0.300i)34-s + (0.764 − 1.84i)37-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.654 + 0.755i)4-s + (0.948 − 1.73i)5-s + (−0.281 − 0.959i)8-s + (−1.58 + 1.18i)10-s + (−0.912 + 0.456i)13-s + (−0.142 + 0.989i)16-s + (−0.398 − 0.148i)17-s + (1.93 − 0.420i)20-s + (−1.57 − 2.45i)25-s + (1.01 − 0.0364i)26-s + (1.13 + 0.121i)29-s + (0.540 − 0.841i)32-s + (0.300 + 0.300i)34-s + (0.764 − 1.84i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $-0.638 + 0.769i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ -0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8267509719\)
\(L(\frac12)\) \(\approx\) \(0.8267509719\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 \)
89 \( 1 + (-0.877 + 0.479i)T \)
good5 \( 1 + (-0.948 + 1.73i)T + (-0.540 - 0.841i)T^{2} \)
7 \( 1 + (-0.212 + 0.977i)T^{2} \)
11 \( 1 + (-0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.912 - 0.456i)T + (0.599 - 0.800i)T^{2} \)
17 \( 1 + (0.398 + 0.148i)T + (0.755 + 0.654i)T^{2} \)
19 \( 1 + (-0.877 - 0.479i)T^{2} \)
23 \( 1 + (-0.479 + 0.877i)T^{2} \)
29 \( 1 + (-1.13 - 0.121i)T + (0.977 + 0.212i)T^{2} \)
31 \( 1 + (0.479 + 0.877i)T^{2} \)
37 \( 1 + (-0.764 + 1.84i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (1.59 + 0.800i)T + (0.599 + 0.800i)T^{2} \)
43 \( 1 + (0.977 - 0.212i)T^{2} \)
47 \( 1 + (-0.989 - 0.142i)T^{2} \)
53 \( 1 + (0.133 + 1.86i)T + (-0.989 + 0.142i)T^{2} \)
59 \( 1 + (0.800 - 0.599i)T^{2} \)
61 \( 1 + (0.276 - 1.53i)T + (-0.936 - 0.349i)T^{2} \)
67 \( 1 + (0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.540 + 0.841i)T^{2} \)
73 \( 1 + (-1.29 - 0.186i)T + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.281 - 0.959i)T^{2} \)
83 \( 1 + (-0.0713 - 0.997i)T^{2} \)
97 \( 1 + (1.91 - 0.562i)T + (0.841 - 0.540i)T^{2} \)
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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

−8.711069968686245641268421250391, −8.207368098607897040806510358051, −7.21418264471558711395525197080, −6.43639951728741544638990901121, −5.45933157380739009204130621324, −4.76013793558607469254882990684, −3.88206824597401354072321545994, −2.41472061782488177807494712688, −1.82887531506320785194448377989, −0.65129167990026552260779823990, 1.60042143013055874164851277194, 2.63499441574324512421294511460, 3.06472122965138197913847275991, 4.73713025257024005612191861109, 5.65645552515230702180700414361, 6.51099409287956042887175366891, 6.68975534992429781885489691063, 7.63077590592619462002587445013, 8.202179579989305445326825230707, 9.347318277364137122277793746509

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