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Properties

Label 2-1449-161.62-c0-0-2
Degree $2$
Conductor $1449$
Sign $-0.847 + 0.530i$
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  + (1.29 − 1.49i)2-s + (−0.415 − 2.88i)4-s + (0.959 − 0.281i)7-s + (−3.19 − 2.05i)8-s + (0.708 + 0.817i)11-s + (0.822 − 1.80i)14-s + (−4.41 + 1.29i)16-s + 2.14·22-s + (−0.909 + 0.415i)23-s + (−0.654 + 0.755i)25-s + (−1.21 − 2.65i)28-s + (−0.215 + 1.49i)29-s + (−2.20 + 4.83i)32-s + (0.797 − 1.74i)37-s + (−0.698 + 0.449i)43-s + (2.06 − 2.38i)44-s + ⋯
L(s)  = 1  + (1.29 − 1.49i)2-s + (−0.415 − 2.88i)4-s + (0.959 − 0.281i)7-s + (−3.19 − 2.05i)8-s + (0.708 + 0.817i)11-s + (0.822 − 1.80i)14-s + (−4.41 + 1.29i)16-s + 2.14·22-s + (−0.909 + 0.415i)23-s + (−0.654 + 0.755i)25-s + (−1.21 − 2.65i)28-s + (−0.215 + 1.49i)29-s + (−2.20 + 4.83i)32-s + (0.797 − 1.74i)37-s + (−0.698 + 0.449i)43-s + (2.06 − 2.38i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.847 + 0.530i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ -0.847 + 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.153418451\)
\(L(\frac12)\) \(\approx\) \(2.153418451\)
\(L(1)\) 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 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (0.909 - 0.415i)T \)
good2 \( 1 + (-1.29 + 1.49i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.708 - 0.817i)T + (-0.142 + 0.989i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.959 - 0.281i)T^{2} \)
29 \( 1 + (0.215 - 1.49i)T + (-0.959 - 0.281i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-0.797 + 1.74i)T + (-0.654 - 0.755i)T^{2} \)
41 \( 1 + (0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.415 - 0.909i)T^{2} \)
67 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.959 - 0.281i)T^{2} \)
79 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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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

−9.611330179283106911357567528233, −9.096381670862450664794402988696, −7.70003461627906894548717585463, −6.64543754603141990139346139701, −5.65828621270108133705196669663, −4.91705503140603866368467055740, −4.15527564681830560478515995598, −3.45640120762692934354579607675, −2.09514010845030981892603209072, −1.44081633520592238657480562654, 2.37638851981547355083490162841, 3.60446593854917578013722853777, 4.36963582372365067086146954375, 5.10374266791867456387398079298, 6.13803035983649047188438039635, 6.37029303405876054451495557087, 7.63605406935818471163552002041, 8.172826366858449640312960562975, 8.679816741413414116427000251015, 9.797718491789657509982066600501

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