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Properties

Label 2-405-135.113-c1-0-5
Degree $2$
Conductor $405$
Sign $0.498 - 0.866i$
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  + (−2.08 − 0.182i)2-s + (2.35 + 0.414i)4-s + (1.98 + 1.03i)5-s + (3.38 + 2.36i)7-s + (−0.784 − 0.210i)8-s + (−3.95 − 2.51i)10-s + (−1.66 + 4.58i)11-s + (−0.223 − 2.55i)13-s + (−6.62 − 5.55i)14-s + (−2.88 − 1.05i)16-s + (2.36 − 0.632i)17-s + (−4.05 + 2.34i)19-s + (4.23 + 3.24i)20-s + (4.32 − 9.26i)22-s + (−2.13 − 3.04i)23-s + ⋯
L(s)  = 1  + (−1.47 − 0.129i)2-s + (1.17 + 0.207i)4-s + (0.887 + 0.461i)5-s + (1.27 + 0.894i)7-s + (−0.277 − 0.0743i)8-s + (−1.24 − 0.795i)10-s + (−0.503 + 1.38i)11-s + (−0.0619 − 0.707i)13-s + (−1.77 − 1.48i)14-s + (−0.722 − 0.262i)16-s + (0.572 − 0.153i)17-s + (−0.930 + 0.537i)19-s + (0.947 + 0.726i)20-s + (0.921 − 1.97i)22-s + (−0.445 − 0.635i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.498 - 0.866i$
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: \((2,\ 405,\ (\ :1/2),\ 0.498 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709196 + 0.410050i\)
\(L(\frac12)\) \(\approx\) \(0.709196 + 0.410050i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.98 - 1.03i)T \)
good2 \( 1 + (2.08 + 0.182i)T + (1.96 + 0.347i)T^{2} \)
7 \( 1 + (-3.38 - 2.36i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (1.66 - 4.58i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.223 + 2.55i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-2.36 + 0.632i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.05 - 2.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.13 + 3.04i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (2.44 - 2.05i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.272 - 1.54i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.592 - 2.21i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.38 + 5.22i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.553 - 1.18i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (-0.220 + 0.314i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (0.177 - 0.177i)T - 53iT^{2} \)
59 \( 1 + (-0.797 + 0.290i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.38 - 7.86i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.09 + 0.708i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-7.83 - 4.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.19 + 11.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.94 + 8.27i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.325 - 3.71i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (7.58 + 13.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.25 - 0.583i)T + (62.3 - 74.3i)T^{2} \)
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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

−10.94862175471008320391190354218, −10.34611801017572638679441481940, −9.664220143576723876866754759661, −8.690634418516386656738182949826, −7.945057941085433313769359330823, −7.11512379125046597486083745829, −5.75942796767878647772050162705, −4.76688185316529411712753745186, −2.44609318984811425890376709279, −1.73784292491883236756955304743, 0.923658218986730734117082146200, 2.08572313219063958829177620738, 4.23305000085811715922196653649, 5.46967881461648687915320216461, 6.63645979537143530327724178136, 7.83659739608219186493190403722, 8.325340956292425760918672674741, 9.226694687783858561078314195285, 10.06632570908010364419521493279, 10.96778965830991777242760888937

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