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Properties

Label 2-1449-161.104-c0-0-2
Degree $2$
Conductor $1449$
Sign $-0.0404 + 0.999i$
Analytic cond. $0.723145$
Root an. cond. $0.850379$
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  + (1.53 − 0.983i)2-s + (0.959 − 2.10i)4-s + (0.654 + 0.755i)7-s + (−0.339 − 2.35i)8-s + (−1.66 − 1.07i)11-s + (1.74 + 0.512i)14-s + (−1.32 − 1.53i)16-s − 3.60·22-s + (0.281 + 0.959i)23-s + (0.841 − 0.540i)25-s + (2.21 − 0.650i)28-s + (0.449 + 0.983i)29-s + (−1.24 − 0.366i)32-s + (−1.25 − 0.368i)37-s + (−0.273 + 1.89i)43-s + (−3.84 + 2.47i)44-s + ⋯
L(s)  = 1  + (1.53 − 0.983i)2-s + (0.959 − 2.10i)4-s + (0.654 + 0.755i)7-s + (−0.339 − 2.35i)8-s + (−1.66 − 1.07i)11-s + (1.74 + 0.512i)14-s + (−1.32 − 1.53i)16-s − 3.60·22-s + (0.281 + 0.959i)23-s + (0.841 − 0.540i)25-s + (2.21 − 0.650i)28-s + (0.449 + 0.983i)29-s + (−1.24 − 0.366i)32-s + (−1.25 − 0.368i)37-s + (−0.273 + 1.89i)43-s + (−3.84 + 2.47i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.0404 + 0.999i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ -0.0404 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.467800666\)
\(L(\frac12)\) \(\approx\) \(2.467800666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
good2 \( 1 + (-1.53 + 0.983i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
11 \( 1 + (1.66 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.449 - 0.983i)T + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (1.25 + 0.368i)T + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.273 - 1.89i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−9.897341145652493076351725299024, −8.720818436601292684148206664666, −7.974668668601301222293155931813, −6.72871875584916099864144572243, −5.63104401161182065234041609626, −5.30691264204745926156229160416, −4.52435659924616196875781414497, −3.17998866827298330353684917418, −2.75447164969598194724844582274, −1.53463004941601476263980009101, 2.21609038507580012304366829361, 3.24633077046094338838834866754, 4.42304037669196521721408109411, 4.85841402223507238245407480582, 5.57089912290411003949333313359, 6.71521994844886008372934577367, 7.32064440191816799671369797833, 7.87558395895168338290140498879, 8.703279503979895574565611104753, 10.23728986640618199906941014628

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