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Properties

Label 2-405-9.7-c3-0-39
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  + (2.22 − 3.86i)2-s + (−5.94 − 10.2i)4-s + (2.5 + 4.33i)5-s + (−2.54 + 4.40i)7-s − 17.3·8-s + 22.2·10-s + (29.1 − 50.4i)11-s + (−10.6 − 18.3i)13-s + (11.3 + 19.6i)14-s + (8.90 − 15.4i)16-s + 68.8·17-s − 40.8·19-s + (29.7 − 51.4i)20-s + (−129. − 225. i)22-s + (−72.1 − 124. i)23-s + ⋯
L(s)  = 1  + (0.788 − 1.36i)2-s + (−0.742 − 1.28i)4-s + (0.223 + 0.387i)5-s + (−0.137 + 0.237i)7-s − 0.765·8-s + 0.705·10-s + (0.799 − 1.38i)11-s + (−0.226 − 0.391i)13-s + (0.216 + 0.374i)14-s + (0.139 − 0.240i)16-s + 0.982·17-s − 0.492·19-s + (0.332 − 0.575i)20-s + (−1.25 − 2.18i)22-s + (−0.654 − 1.13i)23-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

\(L(2)\) \(\approx\) \(2.797354415\)
\(L(\frac12)\) \(\approx\) \(2.797354415\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.22 + 3.86i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (2.54 - 4.40i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-29.1 + 50.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (10.6 + 18.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 68.8T + 4.91e3T^{2} \)
19 \( 1 + 40.8T + 6.85e3T^{2} \)
23 \( 1 + (72.1 + 124. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (110. - 190. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (145. + 252. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 260.T + 5.06e4T^{2} \)
41 \( 1 + (84.8 + 147. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-219. + 379. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (127. - 221. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 214.T + 1.48e5T^{2} \)
59 \( 1 + (-165. - 287. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (27.4 - 47.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (379. + 656. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 904.T + 3.57e5T^{2} \)
73 \( 1 - 866.T + 3.89e5T^{2} \)
79 \( 1 + (103. - 179. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (231. - 401. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 601.T + 7.04e5T^{2} \)
97 \( 1 + (114. - 198. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.80309471782094225096373399643, −9.833895070449593454920141633250, −8.936780042705242729944532024425, −7.68187566459877200943564705994, −6.19480887444077051423470876863, −5.46728304229273889209353715971, −4.04678110281969789333932747345, −3.25611809215888129110920544523, −2.19064690779459661937472753215, −0.73916519564402615524084426520, 1.69199888212212196479062704699, 3.75832023760085803031255485970, 4.57474579152926578248905820544, 5.54373732795504262187738754597, 6.49903397776785556289061693993, 7.31819386621518302806995453663, 8.073068577303326057963764499467, 9.360909117227330697193073983614, 10.00307289666197262537895418387, 11.52876643330547255149436961383

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