Properties

Label 2-405-135.92-c1-0-2
Degree $2$
Conductor $405$
Sign $0.219 - 0.975i$
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  + (−0.160 + 0.0140i)2-s + (−1.94 + 0.342i)4-s + (0.476 − 2.18i)5-s + (−2.37 + 1.66i)7-s + (0.617 − 0.165i)8-s + (−0.0456 + 0.356i)10-s + (1.48 + 4.09i)11-s + (−0.412 + 4.71i)13-s + (0.356 − 0.299i)14-s + (3.61 − 1.31i)16-s + (5.66 + 1.51i)17-s + (0.695 + 0.401i)19-s + (−0.176 + 4.41i)20-s + (−0.296 − 0.635i)22-s + (−1.52 + 2.18i)23-s + ⋯
L(s)  = 1  + (−0.113 + 0.00991i)2-s + (−0.972 + 0.171i)4-s + (0.212 − 0.977i)5-s + (−0.896 + 0.627i)7-s + (0.218 − 0.0585i)8-s + (−0.0144 + 0.112i)10-s + (0.449 + 1.23i)11-s + (−0.114 + 1.30i)13-s + (0.0953 − 0.0800i)14-s + (0.903 − 0.328i)16-s + (1.37 + 0.368i)17-s + (0.159 + 0.0920i)19-s + (−0.0394 + 0.986i)20-s + (−0.0631 − 0.135i)22-s + (−0.318 + 0.455i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.219 - 0.975i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.219 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.646582 + 0.517042i\)
\(L(\frac12)\) \(\approx\) \(0.646582 + 0.517042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.476 + 2.18i)T \)
good2 \( 1 + (0.160 - 0.0140i)T + (1.96 - 0.347i)T^{2} \)
7 \( 1 + (2.37 - 1.66i)T + (2.39 - 6.57i)T^{2} \)
11 \( 1 + (-1.48 - 4.09i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (0.412 - 4.71i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (-5.66 - 1.51i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.695 - 0.401i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.52 - 2.18i)T + (-7.86 - 21.6i)T^{2} \)
29 \( 1 + (2.06 + 1.73i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.27 - 7.23i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.841 - 3.14i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.88 + 3.44i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.90 - 6.22i)T + (-27.6 - 32.9i)T^{2} \)
47 \( 1 + (3.17 + 4.52i)T + (-16.0 + 44.1i)T^{2} \)
53 \( 1 + (-1.70 - 1.70i)T + 53iT^{2} \)
59 \( 1 + (-4.31 - 1.57i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.10 + 11.9i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.65 + 0.582i)T + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (-5.61 + 3.24i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.49 - 9.31i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.322 - 0.384i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.780 - 8.92i)T + (-81.7 + 14.4i)T^{2} \)
89 \( 1 + (-6.84 + 11.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.60 - 3.08i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.92907520047227523909770461918, −9.961123197701585678833010629423, −9.625542388432066927957824214399, −8.920139644466763970508939451887, −7.979414662881680603623343267765, −6.73017359213762821272766035573, −5.51729514472619127944987085046, −4.62323235853229240098812607293, −3.58288404823801596334033927082, −1.61813760210631701019077345371, 0.61305738115205673928502334144, 3.12141847978682974822897515253, 3.75115298587931973406005895761, 5.45603086279410905249743314479, 6.17010048084521429011255991933, 7.41046112467796865700899710586, 8.274718093115253550453149775014, 9.490291589240479603252970069403, 10.12423858420807865382457283738, 10.73630968574037110066622498677

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