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Properties

Label 2-405-45.29-c2-0-39
Degree $2$
Conductor $405$
Sign $-0.998 + 0.0628i$
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  + (0.0484 − 0.0840i)2-s + (1.99 + 3.45i)4-s + (0.121 − 4.99i)5-s + (−8.81 − 5.09i)7-s + 0.775·8-s + (−0.413 − 0.252i)10-s + (−1.54 − 0.894i)11-s + (−11.8 + 6.86i)13-s + (−0.855 + 0.493i)14-s + (−7.94 + 13.7i)16-s − 13.0·17-s − 16.5·19-s + (17.5 − 9.55i)20-s + (−0.150 + 0.0867i)22-s + (20.8 + 36.0i)23-s + ⋯
L(s)  = 1  + (0.0242 − 0.0420i)2-s + (0.498 + 0.863i)4-s + (0.0243 − 0.999i)5-s + (−1.25 − 0.727i)7-s + 0.0968·8-s + (−0.0413 − 0.0252i)10-s + (−0.140 − 0.0812i)11-s + (−0.915 + 0.528i)13-s + (−0.0610 + 0.0352i)14-s + (−0.496 + 0.859i)16-s − 0.769·17-s − 0.872·19-s + (0.875 − 0.477i)20-s + (−0.00682 + 0.00394i)22-s + (0.905 + 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0628i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.998 + 0.0628i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.998 + 0.0628i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00411155 - 0.130655i\)
\(L(\frac12)\) \(\approx\) \(0.00411155 - 0.130655i\)
\(L(2)\) 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 \( 1 + (-0.121 + 4.99i)T \)
good2 \( 1 + (-0.0484 + 0.0840i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (8.81 + 5.09i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (1.54 + 0.894i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (11.8 - 6.86i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 13.0T + 289T^{2} \)
19 \( 1 + 16.5T + 361T^{2} \)
23 \( 1 + (-20.8 - 36.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (38.2 + 22.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (3.23 + 5.59i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 20.6iT - 1.36e3T^{2} \)
41 \( 1 + (-49.3 + 28.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (29.7 + 17.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-30.0 + 52.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 25.5T + 2.80e3T^{2} \)
59 \( 1 + (35.1 - 20.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.6 - 59.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (52.0 - 30.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 91.1iT - 5.04e3T^{2} \)
73 \( 1 - 74.8iT - 5.32e3T^{2} \)
79 \( 1 + (-30.8 + 53.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-29.2 + 50.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 66.7iT - 7.92e3T^{2} \)
97 \( 1 + (-136. - 79.0i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.66044450207147119839051949661, −9.475586585180303492559950444174, −8.934548869529632382760348802719, −7.60333730110272663948287385797, −7.03967777913951128341679446308, −5.86700841454228701221798647789, −4.40975681197714784223183323955, −3.56928648728987878168450908444, −2.12494235793285732388447262808, −0.04955432646734731665078876715, 2.31026621518005410972900244034, 3.03141457148001837486739226691, 4.82213975075437786662769187633, 6.11623543823231267965702369130, 6.53984906503838885027662965872, 7.48563546546412162243827980509, 9.026871934162713691765603909067, 9.784875448293253382246793329162, 10.61766429411384675153882746111, 11.17947634329787096651307567753

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