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Properties

Label 2-405-45.22-c2-0-39
Degree $2$
Conductor $405$
Sign $-0.979 + 0.203i$
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  + (0.5 − 1.86i)2-s + (0.232 + 0.133i)4-s + (−2.5 − 4.33i)5-s + (−3.09 + 11.5i)7-s + (5.83 − 5.83i)8-s + (−9.33 + 2.49i)10-s + (−3.09 − 5.36i)11-s + (−4.30 − 16.0i)13-s + (20.0 + 11.5i)14-s + (−7.42 − 12.8i)16-s + (−22.2 − 22.2i)17-s − 30.5i·19-s − 1.33i·20-s + (−11.5 + 3.09i)22-s + (2.50 + 9.36i)23-s + ⋯
L(s)  = 1  + (0.250 − 0.933i)2-s + (0.0580 + 0.0334i)4-s + (−0.5 − 0.866i)5-s + (−0.442 + 1.65i)7-s + (0.728 − 0.728i)8-s + (−0.933 + 0.249i)10-s + (−0.281 − 0.487i)11-s + (−0.331 − 1.23i)13-s + (1.43 + 0.825i)14-s + (−0.464 − 0.804i)16-s + (−1.31 − 1.31i)17-s − 1.60i·19-s − 0.0669i·20-s + (−0.525 + 0.140i)22-s + (0.109 + 0.407i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.979 + 0.203i$
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: \((2,\ 405,\ (\ :1),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.131302 - 1.27960i\)
\(L(\frac12)\) \(\approx\) \(0.131302 - 1.27960i\)
\(L(2)\) 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 \( 1 + (2.5 + 4.33i)T \)
good2 \( 1 + (-0.5 + 1.86i)T + (-3.46 - 2i)T^{2} \)
7 \( 1 + (3.09 - 11.5i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (3.09 + 5.36i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (4.30 + 16.0i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + (22.2 + 22.2i)T + 289iT^{2} \)
19 \( 1 + 30.5iT - 361T^{2} \)
23 \( 1 + (-2.50 - 9.36i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 + (-0.990 + 0.571i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-2.58 + 4.48i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-16.8 - 16.8i)T + 1.36e3iT^{2} \)
41 \( 1 + (-2.60 + 4.51i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-1.29 - 0.346i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (4.58 - 17.1i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-3.98 + 3.98i)T - 2.80e3iT^{2} \)
59 \( 1 + (33.8 + 19.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.0 - 27.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-46.0 + 12.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 103.T + 5.04e3T^{2} \)
73 \( 1 + (73.2 - 73.2i)T - 5.32e3iT^{2} \)
79 \( 1 + (-83.8 + 48.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (45.5 + 12.2i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 + 18.0iT - 7.92e3T^{2} \)
97 \( 1 + (36.0 - 134. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.04107650228716767585287623735, −9.615731091825396731740295143811, −9.000542525540477543991761267224, −8.025111212249545944633501832728, −6.84314593269163728525287751064, −5.45521118350249632217178480361, −4.66284070177115489446035806065, −3.07824934786688140310244373808, −2.44223281338416046811072796693, −0.48247456626948490408548070858, 1.96938839030454432726272047771, 3.84954784093900852657594931011, 4.47427532409369062143686666112, 6.21368929658699127408501783975, 6.77676453025344222148498688318, 7.43678697923827233477749568896, 8.263855266185105501277565688503, 9.886859336176531967806930504914, 10.62787801565825421690473333721, 11.16283715211363364040703254726

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