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Properties

Label 2-405-45.22-c2-0-22
Degree $2$
Conductor $405$
Sign $0.970 - 0.242i$
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  + (−0.814 + 3.03i)2-s + (−5.10 − 2.94i)4-s + (−4.99 + 0.203i)5-s + (−0.530 + 1.98i)7-s + (4.22 − 4.22i)8-s + (3.44 − 15.3i)10-s + (1.67 + 2.89i)11-s + (−3.82 − 14.2i)13-s + (−5.58 − 3.22i)14-s + (−2.39 − 4.15i)16-s + (−2.65 − 2.65i)17-s − 20.6i·19-s + (26.1 + 13.6i)20-s + (−10.1 + 2.72i)22-s + (6.02 + 22.4i)23-s + ⋯
L(s)  = 1  + (−0.407 + 1.51i)2-s + (−1.27 − 0.737i)4-s + (−0.999 + 0.0407i)5-s + (−0.0757 + 0.282i)7-s + (0.528 − 0.528i)8-s + (0.344 − 1.53i)10-s + (0.152 + 0.263i)11-s + (−0.294 − 1.09i)13-s + (−0.398 − 0.230i)14-s + (−0.149 − 0.259i)16-s + (−0.155 − 0.155i)17-s − 1.08i·19-s + (1.30 + 0.684i)20-s + (−0.462 + 0.123i)22-s + (0.261 + 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.610585 + 0.0752463i\)
\(L(\frac12)\) \(\approx\) \(0.610585 + 0.0752463i\)
\(L(2)\) 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 \( 1 + (4.99 - 0.203i)T \)
good2 \( 1 + (0.814 - 3.03i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (0.530 - 1.98i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.67 - 2.89i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (3.82 + 14.2i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (2.65 + 2.65i)T + 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (-6.02 - 22.4i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (-0.739 + 0.426i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-9.34 + 16.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-38.0 - 38.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (-14.3 + 24.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (30.7 + 8.23i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-7.22 + 26.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-28.6 + 28.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (96.9 + 55.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (47.0 + 81.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-74.9 + 20.0i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 - 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 + (-21.2 + 12.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-28.8 - 7.74i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-5.34 + 19.9i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.08209062754271960660946659834, −9.800709120003306132685519473830, −8.936028517954820372693176107700, −8.050042753540330543639820034261, −7.43082771312427791606759390240, −6.59539584425886617342869225583, −5.45144134355735219624960609372, −4.55797579822303830895792828780, −2.99948782466115868452340185067, −0.36401519979949832026447891143, 1.14158378233546107363186497348, 2.63715773862741152731155332174, 3.84034250556260428468875750024, 4.49134508332764119373547434289, 6.35651214391553320502123412660, 7.54915818695283296568545708909, 8.593160740149478419356090040570, 9.299596798010493252195539189745, 10.39160503453841481533050694533, 10.97784316951135213349251267378

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