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Properties

Label 2-1449-483.251-c0-0-2
Degree $2$
Conductor $1449$
Sign $0.969 - 0.244i$
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  + (0.377 − 0.587i)2-s + (0.212 + 0.465i)4-s + (0.755 + 0.654i)7-s + (1.04 + 0.150i)8-s + (−1.34 + 0.865i)11-s + (0.670 − 0.196i)14-s + (0.148 − 0.171i)16-s + 1.11i·22-s + (−0.479 − 0.877i)23-s + (0.841 + 0.540i)25-s + (−0.144 + 0.490i)28-s + (0.386 + 0.176i)29-s + (0.252 + 0.861i)32-s + (−0.368 − 1.25i)37-s + (−1.89 + 0.273i)43-s + (−0.688 − 0.442i)44-s + ⋯
L(s)  = 1  + (0.377 − 0.587i)2-s + (0.212 + 0.465i)4-s + (0.755 + 0.654i)7-s + (1.04 + 0.150i)8-s + (−1.34 + 0.865i)11-s + (0.670 − 0.196i)14-s + (0.148 − 0.171i)16-s + 1.11i·22-s + (−0.479 − 0.877i)23-s + (0.841 + 0.540i)25-s + (−0.144 + 0.490i)28-s + (0.386 + 0.176i)29-s + (0.252 + 0.861i)32-s + (−0.368 − 1.25i)37-s + (−1.89 + 0.273i)43-s + (−0.688 − 0.442i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.969 - 0.244i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ 0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.497112997\)
\(L(\frac12)\) \(\approx\) \(1.497112997\)
\(L(1)\) 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 \( 1 + (-0.755 - 0.654i)T \)
23 \( 1 + (0.479 + 0.877i)T \)
good2 \( 1 + (-0.377 + 0.587i)T + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (1.34 - 0.865i)T + (0.415 - 0.909i)T^{2} \)
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 \( 1 + (-0.386 - 0.176i)T + (0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.959 + 0.281i)T^{2} \)
37 \( 1 + (0.368 + 1.25i)T + (-0.841 + 0.540i)T^{2} \)
41 \( 1 + (0.841 + 0.540i)T^{2} \)
43 \( 1 + (1.89 - 0.273i)T + (0.959 - 0.281i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.27 + 1.47i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.983 + 1.53i)T + (-0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.647 + 1.00i)T + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 - 0.755i)T^{2} \)
79 \( 1 + (-0.817 + 0.708i)T + (0.142 - 0.989i)T^{2} \)
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

−10.01499312437955844675575832756, −8.813464235510763514645284867220, −8.110747716241556662199741162131, −7.48832431434845281346047485582, −6.56523358256868737618019761582, −5.13640303913892315973726261107, −4.86231694619575043067982972705, −3.62879974665285205359521980503, −2.54068328961178539815629552166, −1.91327631746071066052950018617, 1.21253017409825998423923441324, 2.58079484607653985739766361624, 3.89705527855087285671270540734, 4.98361902927907369162444441251, 5.39553280772720721842874555841, 6.41415864756112685535931240011, 7.20641980857044880023271118728, 7.999135142116873249263247354670, 8.545489734516039598596392250609, 10.01659742383050416706705283502

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