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Properties

Label 2-405-5.3-c2-0-36
Degree $2$
Conductor $405$
Sign $-0.237 + 0.971i$
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  + (2.36 − 2.36i)2-s − 7.14i·4-s + (1.54 + 4.75i)5-s + (6.44 − 6.44i)7-s + (−7.41 − 7.41i)8-s + (14.8 + 7.58i)10-s − 2.04·11-s + (−3.06 − 3.06i)13-s − 30.4i·14-s − 6.44·16-s + (17.1 − 17.1i)17-s − 18.6i·19-s + (33.9 − 11.0i)20-s + (−4.83 + 4.83i)22-s + (8.42 + 8.42i)23-s + ⋯
L(s)  = 1  + (1.18 − 1.18i)2-s − 1.78i·4-s + (0.308 + 0.951i)5-s + (0.920 − 0.920i)7-s + (−0.927 − 0.927i)8-s + (1.48 + 0.758i)10-s − 0.186·11-s + (−0.235 − 0.235i)13-s − 2.17i·14-s − 0.402·16-s + (1.00 − 1.00i)17-s − 0.980i·19-s + (1.69 − 0.550i)20-s + (−0.219 + 0.219i)22-s + (0.366 + 0.366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.237 + 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23215 - 2.84464i\)
\(L(\frac12)\) \(\approx\) \(2.23215 - 2.84464i\)
\(L(2)\) 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 \( 1 + (-1.54 - 4.75i)T \)
good2 \( 1 + (-2.36 + 2.36i)T - 4iT^{2} \)
7 \( 1 + (-6.44 + 6.44i)T - 49iT^{2} \)
11 \( 1 + 2.04T + 121T^{2} \)
13 \( 1 + (3.06 + 3.06i)T + 169iT^{2} \)
17 \( 1 + (-17.1 + 17.1i)T - 289iT^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
23 \( 1 + (-8.42 - 8.42i)T + 529iT^{2} \)
29 \( 1 - 22.5iT - 841T^{2} \)
31 \( 1 + 23.8T + 961T^{2} \)
37 \( 1 + (33.2 - 33.2i)T - 1.36e3iT^{2} \)
41 \( 1 + 4.31T + 1.68e3T^{2} \)
43 \( 1 + (-22.2 - 22.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-0.614 + 0.614i)T - 2.20e3iT^{2} \)
53 \( 1 + (3.33 + 3.33i)T + 2.80e3iT^{2} \)
59 \( 1 - 76.5iT - 3.48e3T^{2} \)
61 \( 1 + 70.7T + 3.72e3T^{2} \)
67 \( 1 + (-15.2 + 15.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 28.6T + 5.04e3T^{2} \)
73 \( 1 + (-48.1 - 48.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 0.232iT - 6.24e3T^{2} \)
83 \( 1 + (-112. - 112. i)T + 6.88e3iT^{2} \)
89 \( 1 - 103. iT - 7.92e3T^{2} \)
97 \( 1 + (51.0 - 51.0i)T - 9.40e3iT^{2} \)
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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

−10.93423744762141342099697988624, −10.38800042480180517654939452923, −9.446302503159999084834047326352, −7.74141837778779136023531094023, −6.93596948609409088986344650579, −5.48127113088104851039590210782, −4.76472705335528900778143747182, −3.52593436190982069577136371429, −2.65623604866697587913790862959, −1.27402500739155377250011633660, 1.86500753477990234913149669574, 3.71990429551734956920944248120, 4.81346550500974995557537983069, 5.52150405479686892613772842039, 6.13081872507587650927025353394, 7.61777508410804248959740771043, 8.221729679907170290771690023732, 9.092676960361564133583187428371, 10.41010896153189992627255406755, 11.88000631294763712696508595611

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