Properties

Label 2-405-9.7-c3-0-12
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: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9763671364\)
\(L(\frac12)\) \(\approx\) \(0.9763671364\)
\(L(\frac{5}{2})\) 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

−11.24217720848927060810332846948, −9.811390665638521358375948076495, −9.077107714747055055248797142505, −8.425052020578346645661421239904, −7.55394054550925843426381606829, −6.50280654379135982273855093085, −5.85201848454111657624298825277, −4.58768982641429185714102578544, −3.11120197277054295144879417692, −1.04106971645300358767896784375, 0.51865190468414100090190551256, 1.85873732817549221747354452230, 3.11680274847789275431131654342, 4.03094424564287645589891679270, 5.63933536785303956169027539964, 6.90738431130613466228408742885, 7.906300266196755379791121207380, 8.936969366337035781244683183477, 9.926298345251979881918537188879, 10.28058560297404711309553622042

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