Properties

Label 2-405-135.113-c1-0-7
Degree $2$
Conductor $405$
Sign $0.253 + 0.967i$
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  + (−1.85 − 0.162i)2-s + (1.46 + 0.257i)4-s + (2.23 + 0.123i)5-s + (−2.56 − 1.79i)7-s + (0.928 + 0.248i)8-s + (−4.13 − 0.592i)10-s + (1.21 − 3.34i)11-s + (0.165 + 1.89i)13-s + (4.47 + 3.75i)14-s + (−4.47 − 1.62i)16-s + (−3.30 + 0.886i)17-s + (5.00 − 2.89i)19-s + (3.23 + 0.756i)20-s + (−2.80 + 6.02i)22-s + (−0.749 − 1.06i)23-s + ⋯
L(s)  = 1  + (−1.31 − 0.115i)2-s + (0.731 + 0.128i)4-s + (0.998 + 0.0552i)5-s + (−0.969 − 0.678i)7-s + (0.328 + 0.0879i)8-s + (−1.30 − 0.187i)10-s + (0.367 − 1.00i)11-s + (0.0460 + 0.525i)13-s + (1.19 + 1.00i)14-s + (−1.11 − 0.407i)16-s + (−0.802 + 0.215i)17-s + (1.14 − 0.663i)19-s + (0.723 + 0.169i)20-s + (−0.598 + 1.28i)22-s + (−0.156 − 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.253 + 0.967i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538601 - 0.415492i\)
\(L(\frac12)\) \(\approx\) \(0.538601 - 0.415492i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.123i)T \)
good2 \( 1 + (1.85 + 0.162i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (2.56 + 1.79i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (-1.21 + 3.34i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.165 - 1.89i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (3.30 - 0.886i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-5.00 + 2.89i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.749 + 1.06i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (0.0144 - 0.0121i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.260 + 1.47i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.61 + 9.77i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.39 + 7.61i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.23 + 6.93i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-6.31 + 9.02i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (3.54 - 3.54i)T - 53iT^{2} \)
59 \( 1 + (-5.27 + 1.91i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.47 - 8.37i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-0.222 + 0.0194i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-0.428 - 0.247i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.07 - 7.75i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.13 - 1.35i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.643 - 7.35i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (4.43 + 7.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.93 + 3.23i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.68658278041718263332090377601, −10.07576041039615085667205552434, −9.111025776406099894793310699477, −8.845230760449240072863448280474, −7.30957208074679464967824696100, −6.66493146078944401861354353472, −5.51140792265407568693116947557, −3.87385235810825155847511428489, −2.31937006180602839406610535859, −0.73095238815773265513842224489, 1.50440156933454808036514484149, 2.88073807420662788219699017632, 4.75296507831197467705657548786, 6.05881741237951936683019132637, 6.84262938741713398108271726620, 7.928896006284284760250674738995, 8.990582664159398782813853566393, 9.754530635723532654897743448142, 9.886774501167027895231197867097, 11.13486092405790604511151753640

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