Properties

Label 2-405-9.7-c3-0-19
Degree $2$
Conductor $405$
Sign $0.939 - 0.342i$
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  + (1 − 1.73i)2-s + (2.00 + 3.46i)4-s + (−2.5 − 4.33i)5-s + 24·8-s − 10·10-s + (−5 + 8.66i)11-s + (40 + 69.2i)13-s + (8.00 − 13.8i)16-s + 7·17-s − 113·19-s + (10 − 17.3i)20-s + (10 + 17.3i)22-s + (40.5 + 70.1i)23-s + (−12.5 + 21.6i)25-s + 160·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 1.06·8-s − 0.316·10-s + (−0.137 + 0.237i)11-s + (0.853 + 1.47i)13-s + (0.125 − 0.216i)16-s + 0.0998·17-s − 1.36·19-s + (0.111 − 0.193i)20-s + (0.0969 + 0.167i)22-s + (0.367 + 0.635i)23-s + (−0.100 + 0.173i)25-s + 1.20·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.475642903\)
\(L(\frac12)\) \(\approx\) \(2.475642903\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (5 - 8.66i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-40 - 69.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 7T + 4.91e3T^{2} \)
19 \( 1 + 113T + 6.85e3T^{2} \)
23 \( 1 + (-40.5 - 70.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-110 + 190. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-94.5 - 163. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 170T + 5.06e4T^{2} \)
41 \( 1 + (-65 - 112. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (80 - 138. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 631T + 1.48e5T^{2} \)
59 \( 1 + (-280 - 484. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (114.5 - 198. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (375 + 649. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 890T + 3.57e5T^{2} \)
73 \( 1 + 890T + 3.89e5T^{2} \)
79 \( 1 + (-13.5 + 23.3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (214.5 - 371. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 750T + 7.04e5T^{2} \)
97 \( 1 + (-740 + 1.28e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.14573202874587909315953686418, −10.19914843527716186449526683077, −8.991720875672659202206114653567, −8.210662595535854973225136133603, −7.13792582509322593793947896495, −6.17178170140818428417941749339, −4.55036436431716144292516809398, −4.00653498524167697162630630571, −2.60246916437341202100463681794, −1.39978303334069136820862239558, 0.797773692822905705911686711929, 2.52283878049274556591214733347, 3.91462561980045135461682604955, 5.16479484681170340555474660403, 6.06817113691749160782675534461, 6.82107133150000345965197748640, 7.908657646158297079983255914482, 8.673071436838610941183490654419, 10.29793437126315969448566650480, 10.57571434005830465618844711512

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