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Properties

Label 2-405-45.29-c2-0-40
Degree $2$
Conductor $405$
Sign $-0.990 - 0.140i$
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  + (1.32 − 2.29i)2-s + (−1.5 − 2.59i)4-s + (2.35 − 4.41i)5-s + (−9.72 − 5.61i)7-s + 2.64·8-s + (−6.99 − 11.2i)10-s + (−3.67 − 2.12i)11-s + (−9.72 + 5.61i)13-s + (−25.7 + 14.8i)14-s + (9.5 − 16.4i)16-s − 10.5·17-s + 20·19-s + (−14.9 + 0.509i)20-s + (−9.72 + 5.61i)22-s + (−2.64 − 4.58i)23-s + ⋯
L(s)  = 1  + (0.661 − 1.14i)2-s + (−0.375 − 0.649i)4-s + (0.470 − 0.882i)5-s + (−1.38 − 0.801i)7-s + 0.330·8-s + (−0.699 − 1.12i)10-s + (−0.334 − 0.192i)11-s + (−0.747 + 0.431i)13-s + (−1.83 + 1.06i)14-s + (0.593 − 1.02i)16-s − 0.622·17-s + 1.05·19-s + (−0.749 + 0.0254i)20-s + (−0.441 + 0.255i)22-s + (−0.115 − 0.199i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.990 - 0.140i$
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: \((2,\ 405,\ (\ :1),\ -0.990 - 0.140i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.129510 + 1.84010i\)
\(L(\frac12)\) \(\approx\) \(0.129510 + 1.84010i\)
\(L(2)\) 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 \( 1 + (-2.35 + 4.41i)T \)
good2 \( 1 + (-1.32 + 2.29i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (9.72 + 5.61i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.67 + 2.12i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.72 - 5.61i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 10.5T + 289T^{2} \)
19 \( 1 - 20T + 361T^{2} \)
23 \( 1 + (2.64 + 4.58i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-7.34 - 4.24i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (13 + 22.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 33.6iT - 1.36e3T^{2} \)
41 \( 1 + (47.7 - 27.5i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-19.4 - 11.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (10.5 - 18.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 84.6T + 2.80e3T^{2} \)
59 \( 1 + (-40.4 + 23.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-11 + 19.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-77.7 + 44.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 67.3iT - 5.32e3T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-37.0 + 64.1i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 89.0iT - 7.92e3T^{2} \)
97 \( 1 + (-19.4 - 11.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.56604295665731126443847518997, −9.829917266277482851429388840096, −9.229568873537425658115857745355, −7.73132264160943604973782032292, −6.66697529048541430050635804106, −5.37551200783241940664741476971, −4.38007629640924290672487617911, −3.39483363129539917061900002433, −2.19670145036657319119577451970, −0.61468472585395227693752407380, 2.45555920698159773163979558018, 3.53946024996240428688171182386, 5.19476214218167018293522644971, 5.82127130538855419506866604897, 6.81235025093920008557401047661, 7.24743682296034173471888101490, 8.622870195850521540476558617329, 9.833242386484401479477884435704, 10.30241479406868555730215201338, 11.68046286992989308686044615359

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