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Properties

Label 2-405-135.83-c1-0-14
Degree $2$
Conductor $405$
Sign $-0.0200 + 0.999i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + (2.12 − 1.48i)2-s + (1.60 − 4.41i)4-s + (2.09 − 0.783i)5-s + (−1.25 + 2.68i)7-s + (−1.80 − 6.75i)8-s + (3.27 − 4.77i)10-s + (1.00 − 1.19i)11-s + (−2.29 + 3.28i)13-s + (1.33 + 7.54i)14-s + (−6.66 − 5.59i)16-s + (−0.490 + 1.82i)17-s + (−2.41 − 1.39i)19-s + (−0.0957 − 10.5i)20-s + (0.352 − 4.02i)22-s + (−6.78 + 3.16i)23-s + ⋯
L(s)  = 1  + (1.49 − 1.04i)2-s + (0.803 − 2.20i)4-s + (0.936 − 0.350i)5-s + (−0.473 + 1.01i)7-s + (−0.639 − 2.38i)8-s + (1.03 − 1.50i)10-s + (0.302 − 0.360i)11-s + (−0.637 + 0.910i)13-s + (0.355 + 2.01i)14-s + (−1.66 − 1.39i)16-s + (−0.118 + 0.443i)17-s + (−0.553 − 0.319i)19-s + (−0.0214 − 2.35i)20-s + (0.0751 − 0.859i)22-s + (−1.41 + 0.659i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0200 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0200 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.0200 + 0.999i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -0.0200 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23650 - 2.28182i\)
\(L(\frac12)\) \(\approx\) \(2.23650 - 2.28182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.09 + 0.783i)T \)
good2 \( 1 + (-2.12 + 1.48i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (1.25 - 2.68i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-1.00 + 1.19i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (2.29 - 3.28i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (0.490 - 1.82i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.41 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.78 - 3.16i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (-0.683 + 3.87i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-5.45 - 1.98i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (0.316 + 0.0847i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.26 - 1.10i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.0509 - 0.582i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-0.690 - 0.321i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (-5.57 + 5.57i)T - 53iT^{2} \)
59 \( 1 + (7.84 - 6.57i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-11.4 + 4.15i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.63 + 1.14i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-6.11 + 3.52i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.93 + 1.05i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (12.9 + 2.28i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.33 - 3.33i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (4.23 - 7.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.95 + 0.696i)T + (95.5 - 16.8i)T^{2} \)
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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

−11.42652560004171928230722347048, −10.18523867238018518877033815467, −9.633104585120140294633401635668, −8.608388670162324944689589925745, −6.53490670896660612738400142153, −5.98407788149714966323331583567, −5.06250852038482325360294651352, −4.02518685965468712386622510208, −2.64584728294453937629331205801, −1.83059841490269804121193592483, 2.55462939363711780091167109132, 3.74700951190684204470126782649, 4.77140642368508578424100710195, 5.80290010612213972571626050680, 6.63626950156627305121631142913, 7.25775844475911186644639299489, 8.317884204248942864441531825122, 9.864769787678342687055415814524, 10.51544672977249727264290289361, 11.93471336273485634342347169384

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