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Properties

Label 2-405-135.124-c1-0-9
Degree $2$
Conductor $405$
Sign $0.889 + 0.456i$
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  + (0.795 − 0.948i)2-s + (0.0810 + 0.459i)4-s + (−2.17 − 0.523i)5-s + (2.24 + 0.395i)7-s + (2.64 + 1.52i)8-s + (−2.22 + 1.64i)10-s + (4.87 − 1.77i)11-s + (0.993 + 1.18i)13-s + (2.15 − 1.81i)14-s + (2.67 − 0.974i)16-s + (−4.61 + 2.66i)17-s + (2.28 − 3.96i)19-s + (0.0645 − 1.04i)20-s + (2.19 − 6.03i)22-s + (5.06 − 0.892i)23-s + ⋯
L(s)  = 1  + (0.562 − 0.670i)2-s + (0.0405 + 0.229i)4-s + (−0.972 − 0.234i)5-s + (0.847 + 0.149i)7-s + (0.935 + 0.539i)8-s + (−0.704 + 0.520i)10-s + (1.46 − 0.534i)11-s + (0.275 + 0.328i)13-s + (0.577 − 0.484i)14-s + (0.669 − 0.243i)16-s + (−1.11 + 0.646i)17-s + (0.524 − 0.909i)19-s + (0.0144 − 0.232i)20-s + (0.468 − 1.28i)22-s + (1.05 − 0.186i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.889 + 0.456i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.889 + 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85954 - 0.448680i\)
\(L(\frac12)\) \(\approx\) \(1.85954 - 0.448680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.523i)T \)
good2 \( 1 + (-0.795 + 0.948i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (-2.24 - 0.395i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-4.87 + 1.77i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.993 - 1.18i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (4.61 - 2.66i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.28 + 3.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.06 + 0.892i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.94 + 4.15i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.228 + 1.29i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (3.18 - 1.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.26 - 2.74i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.88 - 7.92i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.68 + 1.17i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 - 6.64iT - 53T^{2} \)
59 \( 1 + (2.83 + 1.03i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.999 + 5.67i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.22 + 2.65i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.0130 - 0.0226i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.00 + 2.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.6 + 10.6i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-6.96 + 8.29i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.32 - 4.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.48 + 6.81i)T + (-74.3 + 62.3i)T^{2} \)
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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

−11.42751367575906845321137165395, −10.92538915040195023683736959900, −9.147076062819484920807854270636, −8.506928693985844606882551239718, −7.55561681328008551196127320865, −6.49693398665003767867301290700, −4.83391754424733547405934526044, −4.19253811771006865580349971953, −3.19033999413554727324900952350, −1.54974121309177985273257021792, 1.46302168100154330549879882733, 3.65922168068465639392056197953, 4.51380240695798746253389988723, 5.45281147752881404984115302103, 6.88126538860476972854461209074, 7.17928459426766132018068712954, 8.385454620197026859288350679888, 9.404589898805344936279018900558, 10.67785799314659684442626045875, 11.32473900554740515916474727907

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