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Properties

Label 2-405-9.4-c3-0-21
Degree $2$
Conductor $405$
Sign $0.173 + 0.984i$
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  + (−1.36 − 2.36i)2-s + (0.267 − 0.464i)4-s + (−2.5 + 4.33i)5-s + (−3.46 − 6i)7-s − 23.3·8-s + 13.6·10-s + (29.7 + 51.5i)11-s + (12.5 − 21.7i)13-s + (−9.46 + 16.3i)14-s + (29.7 + 51.4i)16-s + 112.·17-s − 122.·19-s + (1.33 + 2.32i)20-s + (81.2 − 140. i)22-s + (48.7 − 84.3i)23-s + ⋯
L(s)  = 1  + (−0.482 − 0.836i)2-s + (0.0334 − 0.0580i)4-s + (−0.223 + 0.387i)5-s + (−0.187 − 0.323i)7-s − 1.03·8-s + 0.431·10-s + (0.815 + 1.41i)11-s + (0.267 − 0.463i)13-s + (−0.180 + 0.312i)14-s + (0.464 + 0.804i)16-s + 1.61·17-s − 1.48·19-s + (0.0149 + 0.0259i)20-s + (0.787 − 1.36i)22-s + (0.441 − 0.764i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.173 + 0.984i$
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: \((2,\ 405,\ (\ :3/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.411656495\)
\(L(\frac12)\) \(\approx\) \(1.411656495\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.36 + 2.36i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (3.46 + 6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-29.7 - 51.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-12.5 + 21.7i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 112.T + 4.91e3T^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 + (-48.7 + 84.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-63.2 - 109. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-54.1 + 93.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 294.T + 5.06e4T^{2} \)
41 \( 1 + (-102. + 178. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-40.0 - 69.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-134. - 233. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 98.7T + 1.48e5T^{2} \)
59 \( 1 + (-152. + 263. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (333. + 577. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-317. + 550. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 826.T + 3.57e5T^{2} \)
73 \( 1 - 751.T + 3.89e5T^{2} \)
79 \( 1 + (11.9 + 20.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (408. + 707. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 513T + 7.04e5T^{2} \)
97 \( 1 + (214. + 370. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.51582205976831638243686793400, −9.965818969714872378517491087729, −9.116218890443540941385585675513, −7.959131515416700656445071694761, −6.85876097421614074254475024440, −6.01184122223669254388450159259, −4.46631950805071963790934046175, −3.28978000979612565290508229440, −2.06883246623794529901977339246, −0.77321721684416384753616876487, 0.928527251024423362074489436654, 2.95461751725381566689677881459, 4.04990952262589589845853524545, 5.74234372994417939970504220329, 6.25668224776226682621264516553, 7.40675335572257749683085743070, 8.395712357023160735237769359965, 8.834493742044508060432981584217, 9.806375236034291186611272795758, 11.17925195123348582607113041576

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