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Properties

Label 2-405-45.43-c2-0-1
Degree $2$
Conductor $405$
Sign $0.383 + 0.923i$
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  + (0.923 + 3.44i)2-s + (−7.56 + 4.36i)4-s + (−2.17 + 4.50i)5-s + (−1.60 − 5.99i)7-s + (−11.9 − 11.9i)8-s + (−17.5 − 3.32i)10-s + (−10.3 + 17.8i)11-s + (1.49 − 5.56i)13-s + (19.1 − 11.0i)14-s + (12.6 − 21.9i)16-s + (3.85 − 3.85i)17-s − 18.4i·19-s + (−3.23 − 43.5i)20-s + (−71.2 − 19.0i)22-s + (5.18 − 19.3i)23-s + ⋯
L(s)  = 1  + (0.461 + 1.72i)2-s + (−1.89 + 1.09i)4-s + (−0.434 + 0.900i)5-s + (−0.229 − 0.856i)7-s + (−1.49 − 1.49i)8-s + (−1.75 − 0.332i)10-s + (−0.939 + 1.62i)11-s + (0.114 − 0.428i)13-s + (1.37 − 0.791i)14-s + (0.791 − 1.37i)16-s + (0.226 − 0.226i)17-s − 0.973i·19-s + (−0.161 − 2.17i)20-s + (−3.23 − 0.867i)22-s + (0.225 − 0.841i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.383 + 0.923i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.383 + 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.456625 - 0.304921i\)
\(L(\frac12)\) \(\approx\) \(0.456625 - 0.304921i\)
\(L(2)\) 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 \( 1 + (2.17 - 4.50i)T \)
good2 \( 1 + (-0.923 - 3.44i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (1.60 + 5.99i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (10.3 - 17.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-1.49 + 5.56i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-3.85 + 3.85i)T - 289iT^{2} \)
19 \( 1 + 18.4iT - 361T^{2} \)
23 \( 1 + (-5.18 + 19.3i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (9.21 + 5.32i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-16.5 - 28.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (35.8 - 35.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (-6.02 - 10.4i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (48.6 - 13.0i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (0.225 + 0.840i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (-11.6 - 11.6i)T + 2.80e3iT^{2} \)
59 \( 1 + (72.8 - 42.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.8 - 60.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-52.5 - 14.0i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 107.T + 5.04e3T^{2} \)
73 \( 1 + (-8.17 - 8.17i)T + 5.32e3iT^{2} \)
79 \( 1 + (63.3 + 36.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-5.72 + 1.53i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 6.20iT - 7.92e3T^{2} \)
97 \( 1 + (20.1 + 75.0i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.04657203011249024932404069302, −10.59534060538761345733711130072, −9.973860032721475210046578399968, −8.593503817512325274516054096783, −7.55196893213361472266460009300, −7.16335902226225854531687248108, −6.43045521176655539902910693504, −5.03507128332291288836520486312, −4.33085912468157606321426122650, −2.98421840389795093127674387913, 0.20295174114854201033047162459, 1.68835493655750148003780066670, 3.08092518916763243663856206140, 3.90348088962384805623157618681, 5.24123487337466654898180232823, 5.80372248480743155701264249297, 7.985656966335806730360993712274, 8.791090824080837516361867244186, 9.517114183644150294271035479444, 10.59592833276536489812596433710

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