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Properties

Label 2-405-9.4-c1-0-0
Degree $2$
Conductor $405$
Sign $0.939 + 0.342i$
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  + (−1.15 − 1.99i)2-s + (−1.65 + 2.86i)4-s + (0.5 − 0.866i)5-s + (1.30 + 2.25i)7-s + 2.99·8-s − 2.30·10-s + (2.30 + 3.98i)11-s + (−3.30 + 5.72i)13-s + (2.99 − 5.19i)14-s + (−0.151 − 0.262i)16-s − 1.60·17-s + 3.60·19-s + (1.65 + 2.86i)20-s + (5.30 − 9.18i)22-s + (−1.5 + 2.59i)23-s + ⋯
L(s)  = 1  + (−0.814 − 1.41i)2-s + (−0.825 + 1.43i)4-s + (0.223 − 0.387i)5-s + (0.492 + 0.852i)7-s + 1.06·8-s − 0.728·10-s + (0.694 + 1.20i)11-s + (−0.916 + 1.58i)13-s + (0.801 − 1.38i)14-s + (−0.0378 − 0.0655i)16-s − 0.389·17-s + 0.827·19-s + (0.369 + 0.639i)20-s + (1.13 − 1.95i)22-s + (−0.312 + 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831585 - 0.146630i\)
\(L(\frac12)\) \(\approx\) \(0.831585 - 0.146630i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.15 + 1.99i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.30 - 2.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.30 - 3.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.30 - 5.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.60T + 17T^{2} \)
19 \( 1 - 3.60T + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.697 - 1.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.80 + 4.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (2.30 - 3.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.302 + 0.524i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.60 + 7.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.60T + 53T^{2} \)
59 \( 1 + (0.697 - 1.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.10 - 3.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.394 + 0.683i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.39T + 71T^{2} \)
73 \( 1 - 12.6T + 73T^{2} \)
79 \( 1 + (-5.80 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.5 - 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.57632355568193106956279577205, −10.07307523798508721675298351165, −9.451272998778866453628407784498, −8.992003331229340422893957781219, −7.86594515876090155904818596983, −6.64433614507061457471715599178, −5.04327687502710730802665851930, −4.05210494209291784399150028066, −2.36316199488495065725759961756, −1.65157750915870385025065854640, 0.76611173884785771570507339475, 3.18423954930775950548306505036, 4.87423338705804572557386282898, 5.87912308039798783819519929362, 6.74348904683304637141656124884, 7.68453296764371841843216192419, 8.232426729069711596972272829467, 9.303631297024531664391205032091, 10.21492317978793203019472246273, 10.90563493464867009040114024881

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