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Properties

Label 2-405-45.7-c2-0-9
Degree $2$
Conductor $405$
Sign $-0.461 - 0.886i$
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  + (2.05 − 0.550i)2-s + (0.459 − 0.265i)4-s + (2.92 + 4.05i)5-s + (−6.15 + 1.64i)7-s + (−5.22 + 5.22i)8-s + (8.24 + 6.72i)10-s + (−7.02 + 12.1i)11-s + (−10.6 − 2.84i)13-s + (−11.7 + 6.78i)14-s + (−8.92 + 15.4i)16-s + (−15.7 − 15.7i)17-s − 15.5i·19-s + (2.42 + 1.08i)20-s + (−7.73 + 28.8i)22-s + (15.4 + 4.12i)23-s + ⋯
L(s)  = 1  + (1.02 − 0.275i)2-s + (0.114 − 0.0663i)4-s + (0.584 + 0.811i)5-s + (−0.879 + 0.235i)7-s + (−0.652 + 0.652i)8-s + (0.824 + 0.672i)10-s + (−0.638 + 1.10i)11-s + (−0.816 − 0.218i)13-s + (−0.838 + 0.484i)14-s + (−0.557 + 0.965i)16-s + (−0.925 − 0.925i)17-s − 0.817i·19-s + (0.121 + 0.0543i)20-s + (−0.351 + 1.31i)22-s + (0.670 + 0.179i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.461 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.461 - 0.886i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.461 - 0.886i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.843808 + 1.39097i\)
\(L(\frac12)\) \(\approx\) \(0.843808 + 1.39097i\)
\(L(2)\) 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 \( 1 + (-2.92 - 4.05i)T \)
good2 \( 1 + (-2.05 + 0.550i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (6.15 - 1.64i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (7.02 - 12.1i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (10.6 + 2.84i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (15.7 + 15.7i)T + 289iT^{2} \)
19 \( 1 + 15.5iT - 361T^{2} \)
23 \( 1 + (-15.4 - 4.12i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-29.8 - 17.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-24.7 - 42.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-22.8 - 22.8i)T + 1.36e3iT^{2} \)
41 \( 1 + (-0.283 - 0.490i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (0.142 + 0.533i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-31.3 + 8.41i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-45.4 + 45.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (39.4 - 22.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.18 - 10.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.4 - 117. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 48.1T + 5.04e3T^{2} \)
73 \( 1 + (77.4 - 77.4i)T - 5.32e3iT^{2} \)
79 \( 1 + (-114. - 66.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (25.3 + 94.4i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (-71.5 + 19.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.55106329534372514402328012703, −10.45549219050658343956411379753, −9.686528522830697693406941404994, −8.796852831617582505636576275325, −7.18753026047134346597393156080, −6.58915746963171696419692997674, −5.28998758816917788231427640711, −4.59580822107486295804537956658, −2.93230315935757701269266069838, −2.57158815425841051627525794726, 0.47299879389982125923278002293, 2.62343624580494375648232782001, 3.95813095928131327625606928487, 4.87070877768196924302540885323, 5.97181749667619915104899836236, 6.38846250467280802423141007915, 7.946490497605345836299500804697, 9.026535584476093376201330082773, 9.767094338619575413166147912015, 10.68990346472796431706758017450

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