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Properties

Label 2-4410-105.104-c1-0-0
Degree $2$
Conductor $4410$
Sign $-0.846 + 0.531i$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−0.526 + 2.17i)5-s − 8-s + (0.526 − 2.17i)10-s − 4.41i·11-s − 4.94·13-s + 16-s + 5.22i·17-s + 4.97i·19-s + (−0.526 + 2.17i)20-s + 4.41i·22-s + 7.71·23-s + (−4.44 − 2.28i)25-s + 4.94·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.235 + 0.971i)5-s − 0.353·8-s + (0.166 − 0.687i)10-s − 1.33i·11-s − 1.37·13-s + 0.250·16-s + 1.26i·17-s + 1.14i·19-s + (−0.117 + 0.485i)20-s + 0.942i·22-s + 1.60·23-s + (−0.889 − 0.457i)25-s + 0.969·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.846 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.846 + 0.531i$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4410} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -0.846 + 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04129923884\)
\(L(\frac12)\) \(\approx\) \(0.04129923884\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + (0.526 - 2.17i)T \)
7 \( 1 \)
good11 \( 1 + 4.41iT - 11T^{2} \)
13 \( 1 + 4.94T + 13T^{2} \)
17 \( 1 - 5.22iT - 17T^{2} \)
19 \( 1 - 4.97iT - 19T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 - 6.43iT - 31T^{2} \)
37 \( 1 - 9.25iT - 37T^{2} \)
41 \( 1 - 6.79T + 41T^{2} \)
43 \( 1 - 2.08iT - 43T^{2} \)
47 \( 1 + 7.83iT - 47T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 6.76iT - 61T^{2} \)
67 \( 1 + 2.40iT - 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + 0.202T + 73T^{2} \)
79 \( 1 + 0.736T + 79T^{2} \)
83 \( 1 - 3.96iT - 83T^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−8.594157643042943768004612821847, −8.071097637113591217641045359420, −7.50118118852598415783682436033, −6.59630209918640208109346119254, −6.15676035029649818511177487409, −5.26242048447979744496691991333, −4.09277471403886782470027127752, −3.19673932487734913240107168238, −2.60652304174612880943381518750, −1.40629748000445359275720317098, 0.01609045634541849569488889516, 1.10794231542790941277625347089, 2.25503880573317006303804873523, 2.99476584306199595041367221228, 4.49292405364502718343408587974, 4.82311367539754858358354145279, 5.57206973061649364429067163803, 6.94335210484610146397843369581, 7.28547721913058935389641892900, 7.76336925163381832928441049203

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