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Properties

Label 2-405-9.4-c3-0-31
Degree $2$
Conductor $405$
Sign $0.766 - 0.642i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−3.99 + 6.92i)4-s + (2.5 − 4.33i)5-s + (−3 − 5.19i)7-s + 20·10-s + (−16 − 27.7i)11-s + (19 − 32.9i)13-s + (12 − 20.7i)14-s + (31.9 + 55.4i)16-s + 26·17-s + 100·19-s + (20.0 + 34.6i)20-s + (63.9 − 110. i)22-s + (39 − 67.5i)23-s + (−12.5 − 21.6i)25-s + 152·26-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.223 − 0.387i)5-s + (−0.161 − 0.280i)7-s + 0.632·10-s + (−0.438 − 0.759i)11-s + (0.405 − 0.702i)13-s + (0.229 − 0.396i)14-s + (0.499 + 0.866i)16-s + 0.370·17-s + 1.20·19-s + (0.223 + 0.387i)20-s + (0.620 − 1.07i)22-s + (0.353 − 0.612i)23-s + (−0.100 − 0.173i)25-s + 1.14·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.050824654\)
\(L(\frac12)\) \(\approx\) \(3.050824654\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (-2 - 3.46i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (3 + 5.19i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (16 + 27.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-19 + 32.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 26T + 4.91e3T^{2} \)
19 \( 1 - 100T + 6.85e3T^{2} \)
23 \( 1 + (-39 + 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-25 - 43.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-54 + 93.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 266T + 5.06e4T^{2} \)
41 \( 1 + (11 - 19.0i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (221 + 382. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-257 - 445. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 2T + 1.48e5T^{2} \)
59 \( 1 + (250 - 433. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-259 - 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (63 - 109. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 412T + 3.57e5T^{2} \)
73 \( 1 + 878T + 3.89e5T^{2} \)
79 \( 1 + (300 + 519. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (141 + 244. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 150T + 7.04e5T^{2} \)
97 \( 1 + (193 + 334. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.86951685656738344899331292932, −10.03132147853888873729213535457, −8.758482121553877006308497796568, −7.929465598469521115816880658222, −7.11448364840591370913942027896, −5.93520628191959503933118557969, −5.44060499498308715617881966249, −4.30978486393932387358456080192, −3.06483354527067517422971676522, −0.940954114857839875114289992978, 1.34701853984755522017106908753, 2.52856591930502184954192232396, 3.46191836835597339796134806840, 4.63109012096907146754365110452, 5.59704928133762966125237090968, 6.90887239940080653372790351578, 7.911648146412266472245248909525, 9.430819919916220368927843657418, 9.957009281608892151805291097542, 10.96564691004052362523405092299

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