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Properties

Label 2-405-135.83-c1-0-4
Degree $2$
Conductor $405$
Sign $-0.451 - 0.892i$
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  + (−2.05 + 1.43i)2-s + (1.46 − 4.03i)4-s + (1.82 + 1.29i)5-s + (−1.14 + 2.44i)7-s + (1.48 + 5.55i)8-s + (−5.60 − 0.0236i)10-s + (2.09 − 2.50i)11-s + (1.80 − 2.57i)13-s + (−1.17 − 6.67i)14-s + (−4.47 − 3.75i)16-s + (−1.57 + 5.88i)17-s + (2.91 + 1.68i)19-s + (7.88 − 5.47i)20-s + (−0.714 + 8.16i)22-s + (4.50 − 2.10i)23-s + ⋯
L(s)  = 1  + (−1.45 + 1.01i)2-s + (0.734 − 2.01i)4-s + (0.816 + 0.577i)5-s + (−0.431 + 0.925i)7-s + (0.526 + 1.96i)8-s + (−1.77 − 0.00747i)10-s + (0.633 − 0.754i)11-s + (0.500 − 0.714i)13-s + (−0.314 − 1.78i)14-s + (−1.11 − 0.939i)16-s + (−0.382 + 1.42i)17-s + (0.669 + 0.386i)19-s + (1.76 − 1.22i)20-s + (−0.152 + 1.74i)22-s + (0.939 − 0.437i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.451 - 0.892i$
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: \((2,\ 405,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393222 + 0.639412i\)
\(L(\frac12)\) \(\approx\) \(0.393222 + 0.639412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.29i)T \)
good2 \( 1 + (2.05 - 1.43i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (1.14 - 2.44i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-2.09 + 2.50i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.80 + 2.57i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (1.57 - 5.88i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.91 - 1.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.50 + 2.10i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (0.541 - 3.07i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.346 - 0.125i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (4.28 + 1.14i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.54 - 1.33i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.386 - 4.42i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-6.23 - 2.90i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (3.48 - 3.48i)T - 53iT^{2} \)
59 \( 1 + (5.87 - 4.92i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (6.73 - 2.45i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.315 + 0.221i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-8.46 + 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0248 - 0.00667i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.9 - 1.93i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.627 - 0.896i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-3.93 + 6.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.10 - 0.446i)T + (95.5 - 16.8i)T^{2} \)
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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

−10.90712261165181405273236434802, −10.50242548059245244805907340961, −9.377898598136679826648174885125, −8.884342836524266461442416451457, −8.045010000476743900326055359060, −6.79136145404870956578406205066, −6.11671914127825424271058728006, −5.54512374601508392103961313854, −3.15356067045822445817816653651, −1.49171922871098973201352554945, 0.878047935829197820625564541801, 2.07241750707596354111081393519, 3.51005682659101217982238814944, 4.88820877977862317998083771708, 6.72966733092979732270705691938, 7.33467176776534319921736710424, 8.706300201273230203217110554073, 9.404648826102470695886573835023, 9.795852456471160786014797392978, 10.77822048805370093226726266044

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