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Properties

Label 2-405-45.43-c2-0-15
Degree $2$
Conductor $405$
Sign $-0.356 + 0.934i$
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  + (−1.01 − 3.79i)2-s + (−9.91 + 5.72i)4-s + (−3.87 − 3.15i)5-s + (2.11 + 7.89i)7-s + (20.7 + 20.7i)8-s + (−8.04 + 17.9i)10-s + (1.25 − 2.17i)11-s + (−0.784 + 2.92i)13-s + (27.8 − 16.0i)14-s + (34.6 − 60.0i)16-s + (8.93 − 8.93i)17-s − 6.84i·19-s + (56.5 + 9.10i)20-s + (−9.53 − 2.55i)22-s + (−2.63 + 9.82i)23-s + ⋯
L(s)  = 1  + (−0.508 − 1.89i)2-s + (−2.47 + 1.43i)4-s + (−0.775 − 0.631i)5-s + (0.302 + 1.12i)7-s + (2.58 + 2.58i)8-s + (−0.804 + 1.79i)10-s + (0.114 − 0.197i)11-s + (−0.0603 + 0.225i)13-s + (1.98 − 1.14i)14-s + (2.16 − 3.75i)16-s + (0.525 − 0.525i)17-s − 0.360i·19-s + (2.82 + 0.455i)20-s + (−0.433 − 0.116i)22-s + (−0.114 + 0.427i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.356 + 0.934i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.356 + 0.934i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.512103 - 0.743276i\)
\(L(\frac12)\) \(\approx\) \(0.512103 - 0.743276i\)
\(L(2)\) 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 \( 1 + (3.87 + 3.15i)T \)
good2 \( 1 + (1.01 + 3.79i)T + (-3.46 + 2i)T^{2} \)
7 \( 1 + (-2.11 - 7.89i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.25 + 2.17i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (0.784 - 2.92i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-8.93 + 8.93i)T - 289iT^{2} \)
19 \( 1 + 6.84iT - 361T^{2} \)
23 \( 1 + (2.63 - 9.82i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (0.184 + 0.106i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.32 - 10.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-32.7 + 32.7i)T - 1.36e3iT^{2} \)
41 \( 1 + (-30.7 - 53.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-44.9 + 12.0i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (8.21 + 30.6i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (37.7 + 37.7i)T + 2.80e3iT^{2} \)
59 \( 1 + (-79.6 + 46.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (26.1 - 45.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (75.3 + 20.1i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 66.1T + 5.04e3T^{2} \)
73 \( 1 + (-71.8 - 71.8i)T + 5.32e3iT^{2} \)
79 \( 1 + (-104. - 60.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-29.9 + 8.03i)T + (5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + (-14.9 - 55.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.16206892164073835048707756649, −9.792244254737020662120433779108, −9.133661969064995916024356183354, −8.439958880338269294557694830086, −7.63434506693401177882737049029, −5.39562033430372386960254067144, −4.48683173756446975093928771004, −3.37634225848337928817784661131, −2.23682370676985838995815115504, −0.823877433980470587993563006719, 0.77805022734125397402472566747, 3.85640343284224391766190553536, 4.61283107850114430733357152357, 5.97156597816246001669465062173, 6.80693683901566891605465750894, 7.76216502915828137948585987677, 7.935361398865374011664995785258, 9.242357313947641098396378168158, 10.27115038420442988919061393923, 10.82686312631011709190021426917

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