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Properties

Label 2-405-5.2-c2-0-37
Degree $2$
Conductor $405$
Sign $-0.340 + 0.940i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.456 + 0.456i)2-s − 3.58i·4-s + (3.92 − 3.10i)5-s + (−4.20 − 4.20i)7-s + (3.46 − 3.46i)8-s + (3.20 + 0.373i)10-s − 0.0953·11-s + (2.41 − 2.41i)13-s − 3.84i·14-s − 11.1·16-s + (−14.4 − 14.4i)17-s + 25.7i·19-s + (−11.1 − 14.0i)20-s + (−0.0435 − 0.0435i)22-s + (−3.36 + 3.36i)23-s + ⋯
L(s)  = 1  + (0.228 + 0.228i)2-s − 0.895i·4-s + (0.784 − 0.620i)5-s + (−0.601 − 0.601i)7-s + (0.433 − 0.433i)8-s + (0.320 + 0.0373i)10-s − 0.00866·11-s + (0.185 − 0.185i)13-s − 0.274i·14-s − 0.697·16-s + (−0.847 − 0.847i)17-s + 1.35i·19-s + (−0.555 − 0.702i)20-s + (−0.00198 − 0.00198i)22-s + (−0.146 + 0.146i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.340 + 0.940i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.340 + 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00214 - 1.42931i\)
\(L(\frac12)\) \(\approx\) \(1.00214 - 1.42931i\)
\(L(2)\) 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 \( 1 + (-3.92 + 3.10i)T \)
good2 \( 1 + (-0.456 - 0.456i)T + 4iT^{2} \)
7 \( 1 + (4.20 + 4.20i)T + 49iT^{2} \)
11 \( 1 + 0.0953T + 121T^{2} \)
13 \( 1 + (-2.41 + 2.41i)T - 169iT^{2} \)
17 \( 1 + (14.4 + 14.4i)T + 289iT^{2} \)
19 \( 1 - 25.7iT - 361T^{2} \)
23 \( 1 + (3.36 - 3.36i)T - 529iT^{2} \)
29 \( 1 + 41.1iT - 841T^{2} \)
31 \( 1 + 37T + 961T^{2} \)
37 \( 1 + (-50.6 - 50.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 2.26T + 1.68e3T^{2} \)
43 \( 1 + (-24.7 + 24.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-16.3 - 16.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (-49.0 + 49.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 100. iT - 3.48e3T^{2} \)
61 \( 1 + 55.4T + 3.72e3T^{2} \)
67 \( 1 + (-37.3 - 37.3i)T + 4.48e3iT^{2} \)
71 \( 1 - 39.8T + 5.04e3T^{2} \)
73 \( 1 + (-39.3 + 39.3i)T - 5.32e3iT^{2} \)
79 \( 1 - 60.0iT - 6.24e3T^{2} \)
83 \( 1 + (-77.3 + 77.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 51.6iT - 7.92e3T^{2} \)
97 \( 1 + (-70.7 - 70.7i)T + 9.40e3iT^{2} \)
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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

−10.52925412514265697254272906517, −9.816132543176288894807475987421, −9.253142087847148406567507224522, −7.935335237246986844160981725367, −6.67653685290306673647801642586, −5.98152699382803798622707358581, −5.04507769173012803947475836641, −3.95017223044317728241588928044, −2.10071044125648832011973784700, −0.68229501368343039056926341271, 2.17385623250824332003935417820, 3.02252416303470129770793467437, 4.24323431719983969590345650831, 5.63229801592760774581258984864, 6.62911246631826056232888350331, 7.44438618448966483719484649689, 8.893330411875019482883604764598, 9.234359997660244938738973949154, 10.72629187052173667882515333658, 11.14963849471781800170635604373

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