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Properties

Label 2-405-5.4-c3-0-16
Degree $2$
Conductor $405$
Sign $0.749 + 0.661i$
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  − 4.80i·2-s − 15.0·4-s + (8.38 + 7.40i)5-s − 5.92i·7-s + 33.9i·8-s + (35.5 − 40.2i)10-s − 40.2·11-s + 42.1i·13-s − 28.4·14-s + 42.6·16-s + 93.7i·17-s + 132.·19-s + (−126. − 111. i)20-s + 193. i·22-s − 45.1i·23-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.88·4-s + (0.749 + 0.661i)5-s − 0.319i·7-s + 1.50i·8-s + (1.12 − 1.27i)10-s − 1.10·11-s + 0.898i·13-s − 0.543·14-s + 0.665·16-s + 1.33i·17-s + 1.59·19-s + (−1.41 − 1.24i)20-s + 1.87i·22-s − 0.409i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.749 + 0.661i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.607797765\)
\(L(\frac12)\) \(\approx\) \(1.607797765\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.38 - 7.40i)T \)
good2 \( 1 + 4.80iT - 8T^{2} \)
7 \( 1 + 5.92iT - 343T^{2} \)
11 \( 1 + 40.2T + 1.33e3T^{2} \)
13 \( 1 - 42.1iT - 2.19e3T^{2} \)
17 \( 1 - 93.7iT - 4.91e3T^{2} \)
19 \( 1 - 132.T + 6.85e3T^{2} \)
23 \( 1 + 45.1iT - 1.21e4T^{2} \)
29 \( 1 - 68.8T + 2.43e4T^{2} \)
31 \( 1 + 3.17T + 2.97e4T^{2} \)
37 \( 1 - 386. iT - 5.06e4T^{2} \)
41 \( 1 - 483.T + 6.89e4T^{2} \)
43 \( 1 + 22.0iT - 7.95e4T^{2} \)
47 \( 1 + 96.2iT - 1.03e5T^{2} \)
53 \( 1 + 487. iT - 1.48e5T^{2} \)
59 \( 1 + 783.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 - 572. iT - 3.00e5T^{2} \)
71 \( 1 + 771.T + 3.57e5T^{2} \)
73 \( 1 - 651. iT - 3.89e5T^{2} \)
79 \( 1 - 763.T + 4.93e5T^{2} \)
83 \( 1 - 741. iT - 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 - 537. iT - 9.12e5T^{2} \)
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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

−10.70481437389173137391524176012, −10.08335294267255128356345109136, −9.435881569758126006945432576353, −8.266624603826611964320354196746, −6.97779695996373034480268235570, −5.68764802399142215389193396681, −4.47857170533493914969227435335, −3.28411726899124271814128442669, −2.37634508984862495310937146131, −1.24450236169028076514505266676, 0.59963235571003380309756552486, 2.75861241526462073047950036585, 4.70718260120564965421419367568, 5.46012466124545432631516517863, 5.92197234358507008172364458135, 7.43707974300903747464196300632, 7.79147330402003869218300837533, 9.075272781293819539100570612057, 9.476372368653276533325833376528, 10.68325685543219727693859347354

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