Properties

Label 2-405-45.4-c1-0-11
Degree $2$
Conductor $405$
Sign $0.884 - 0.467i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.686 + 0.396i)2-s + (−0.686 + 1.18i)4-s + (2.18 + 0.469i)5-s + (3 − 1.73i)7-s − 2.67i·8-s + (−1.68 + 0.543i)10-s + (−2.18 − 3.78i)11-s + (5.05 + 2.92i)13-s + (−1.37 + 2.37i)14-s + (−0.313 − 0.543i)16-s − 0.792i·17-s + 0.372·19-s + (−2.05 + 2.27i)20-s + (3 + 1.73i)22-s + (−1.37 − 0.792i)23-s + ⋯
L(s)  = 1  + (−0.485 + 0.280i)2-s + (−0.343 + 0.594i)4-s + (0.977 + 0.210i)5-s + (1.13 − 0.654i)7-s − 0.944i·8-s + (−0.533 + 0.171i)10-s + (−0.659 − 1.14i)11-s + (1.40 + 0.809i)13-s + (−0.366 + 0.635i)14-s + (−0.0784 − 0.135i)16-s − 0.192i·17-s + 0.0854·19-s + (−0.460 + 0.508i)20-s + (0.639 + 0.369i)22-s + (−0.286 − 0.165i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.884 - 0.467i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.884 - 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25907 + 0.312401i\)
\(L(\frac12)\) \(\approx\) \(1.25907 + 0.312401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.469i)T \)
good2 \( 1 + (0.686 - 0.396i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.05 - 2.92i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.792iT - 17T^{2} \)
19 \( 1 - 0.372T + 19T^{2} \)
23 \( 1 + (1.37 + 0.792i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.87 - 4.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.18 - 5.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.37iT - 37T^{2} \)
41 \( 1 + (-2.18 + 3.78i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.62 - 0.939i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.9iT - 53T^{2} \)
59 \( 1 + (0.813 - 1.40i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.68 + 8.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 2.37iT - 73T^{2} \)
79 \( 1 + (-3.37 - 5.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.3 + 5.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (1.11 - 0.644i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.87880732439783344899916516461, −10.66488541651082045642772597857, −9.184084483784814404489487193392, −8.652347980830813372491826750920, −7.75948769102084521591867248910, −6.75458117853901489325220944559, −5.66961564323328366589907725148, −4.42817926571289451701335508230, −3.19033413445418992564815279082, −1.34904551419603019897360392249, 1.41886856847959031644662855358, 2.37701118548688687221684422504, 4.55470365631052148218914378790, 5.44511324009026278280132152031, 6.10662732087322543181469844625, 7.87336905242876427018248083427, 8.536552790442793461724505051866, 9.466218239627373601097735335659, 10.23237524137159125663743426387, 10.93467159669938905150065030982

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