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Properties

Label 2-1449-161.41-c0-0-2
Degree $2$
Conductor $1449$
Sign $-0.853 - 0.521i$
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  + (−1.03 − 0.304i)2-s + (0.142 + 0.0914i)4-s + (−0.415 − 0.909i)7-s + (0.588 + 0.678i)8-s + (−1.45 + 0.425i)11-s + (0.153 + 1.07i)14-s + (−0.473 − 1.03i)16-s + 1.63·22-s + (−0.989 − 0.142i)23-s + (−0.959 − 0.281i)25-s + (0.0240 − 0.167i)28-s + (−0.474 + 0.304i)29-s + (0.0476 + 0.331i)32-s + (0.118 + 0.822i)37-s + (−0.186 + 0.215i)43-s + (−0.245 − 0.0720i)44-s + ⋯
L(s)  = 1  + (−1.03 − 0.304i)2-s + (0.142 + 0.0914i)4-s + (−0.415 − 0.909i)7-s + (0.588 + 0.678i)8-s + (−1.45 + 0.425i)11-s + (0.153 + 1.07i)14-s + (−0.473 − 1.03i)16-s + 1.63·22-s + (−0.989 − 0.142i)23-s + (−0.959 − 0.281i)25-s + (0.0240 − 0.167i)28-s + (−0.474 + 0.304i)29-s + (0.0476 + 0.331i)32-s + (0.118 + 0.822i)37-s + (−0.186 + 0.215i)43-s + (−0.245 − 0.0720i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $-0.853 - 0.521i$
Analytic conductor: \(0.723145\)
Root analytic conductor: \(0.850379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1449,\ (\ :0),\ -0.853 - 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03006366048\)
\(L(\frac12)\) \(\approx\) \(0.03006366048\)
\(L(1)\) 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 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (0.989 + 0.142i)T \)
good2 \( 1 + (1.03 + 0.304i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (1.45 - 0.425i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (-0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.474 - 0.304i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.186 - 0.215i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.234 + 0.512i)T + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.959 + 0.281i)T^{2} \)
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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

−9.493433704082314390151566647500, −8.391944120846472437945397523919, −7.80827876363633104180401151782, −7.19849779177903936790023291673, −6.01256127496400073983898191277, −5.00086096364896169800371093294, −4.12604227243324779367961810532, −2.80947616711092656885650283829, −1.66950623588248696767062959021, −0.03313446554738286266825011896, 1.96865568089082098152751902992, 3.11578924776037719451219499715, 4.28118801315272421879521827886, 5.52336478813218017852600192686, 6.08926929532092546672969399280, 7.33018503566409022293831431008, 7.88205908790177990550477349042, 8.600660172774070479539796500496, 9.288154683128845888330848571347, 10.01270810454854900871816190404

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