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Properties

Label 2-3204-1068.359-c0-0-1
Degree $2$
Conductor $3204$
Sign $0.755 + 0.655i$
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  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (0.905 + 1.21i)5-s + (−0.989 − 0.142i)8-s + (1.50 − 0.107i)10-s + (0.610 + 0.655i)13-s + (−0.654 + 0.755i)16-s + (0.373 − 1.71i)17-s + (0.724 − 1.32i)20-s + (−0.361 + 1.23i)25-s + (0.881 − 0.158i)26-s + (0.183 + 0.109i)29-s + (0.281 + 0.959i)32-s + (−1.24 − 1.24i)34-s + (1.81 + 0.753i)37-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (0.905 + 1.21i)5-s + (−0.989 − 0.142i)8-s + (1.50 − 0.107i)10-s + (0.610 + 0.655i)13-s + (−0.654 + 0.755i)16-s + (0.373 − 1.71i)17-s + (0.724 − 1.32i)20-s + (−0.361 + 1.23i)25-s + (0.881 − 0.158i)26-s + (0.183 + 0.109i)29-s + (0.281 + 0.959i)32-s + (−1.24 − 1.24i)34-s + (1.81 + 0.753i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3204\)    =    \(2^{2} \cdot 3^{2} \cdot 89\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(1.59900\)
Root analytic conductor: \(1.26451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3204} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3204,\ (\ :0),\ 0.755 + 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897732118\)
\(L(\frac12)\) \(\approx\) \(1.897732118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 + 0.841i)T \)
3 \( 1 \)
89 \( 1 + (0.800 + 0.599i)T \)
good5 \( 1 + (-0.905 - 1.21i)T + (-0.281 + 0.959i)T^{2} \)
7 \( 1 + (-0.877 + 0.479i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.610 - 0.655i)T + (-0.0713 + 0.997i)T^{2} \)
17 \( 1 + (-0.373 + 1.71i)T + (-0.909 - 0.415i)T^{2} \)
19 \( 1 + (0.800 - 0.599i)T^{2} \)
23 \( 1 + (-0.599 - 0.800i)T^{2} \)
29 \( 1 + (-0.183 - 0.109i)T + (0.479 + 0.877i)T^{2} \)
31 \( 1 + (0.599 - 0.800i)T^{2} \)
37 \( 1 + (-1.81 - 0.753i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.928 - 0.997i)T + (-0.0713 - 0.997i)T^{2} \)
43 \( 1 + (0.479 - 0.877i)T^{2} \)
47 \( 1 + (-0.755 - 0.654i)T^{2} \)
53 \( 1 + (-0.148 - 0.398i)T + (-0.755 + 0.654i)T^{2} \)
59 \( 1 + (-0.997 + 0.0713i)T^{2} \)
61 \( 1 + (-1.50 + 1.21i)T + (0.212 - 0.977i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.281 - 0.959i)T^{2} \)
73 \( 1 + (0.627 + 0.544i)T + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.989 - 0.142i)T^{2} \)
83 \( 1 + (-0.349 - 0.936i)T^{2} \)
97 \( 1 + (0.266 - 1.85i)T + (-0.959 - 0.281i)T^{2} \)
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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

−9.190306187189758920880489233683, −8.054978541383570711787552043036, −6.88125465627670037137869614748, −6.50377139479394739440284159292, −5.65431263003525143546369430196, −4.90673304784657360260342113845, −3.90519322068123362430686348245, −2.92498802813438917948106051184, −2.47202428435657048726624240680, −1.30388760077839179780378794379, 1.20588247267754258619604068578, 2.47375158258991812598094119226, 3.75887556355572554959240766588, 4.35071476391316458220303539859, 5.48294217580497736916931205701, 5.67890267909895910224410982971, 6.42437013497291988318308494988, 7.45365199260709808475174833154, 8.430641299377378742247000241488, 8.519857517662735942655288730132

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