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Properties

Label 2-1620-45.29-c2-0-18
Degree $2$
Conductor $1620$
Sign $0.691 - 0.722i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.21 − 4.48i)5-s + (8.02 + 4.63i)7-s + (13.2 + 7.64i)11-s + (2.59 − 1.5i)13-s + 4.00·17-s − 27.7·19-s + (18.9 + 32.7i)23-s + (−15.2 + 19.8i)25-s + (39.7 + 22.9i)29-s + (−14.3 − 24.9i)31-s + (3.02 − 46.2i)35-s + 34.3i·37-s + (−23.6 + 13.6i)41-s + (−31.4 − 18.1i)43-s + (−10.0 + 17.3i)47-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)5-s + (1.14 + 0.661i)7-s + (1.20 + 0.695i)11-s + (0.199 − 0.115i)13-s + 0.235·17-s − 1.46·19-s + (0.823 + 1.42i)23-s + (−0.608 + 0.793i)25-s + (1.37 + 0.791i)29-s + (−0.464 − 0.804i)31-s + (0.0865 − 1.32i)35-s + 0.927i·37-s + (−0.577 + 0.333i)41-s + (−0.730 − 0.421i)43-s + (−0.213 + 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.117999548\)
\(L(\frac12)\) \(\approx\) \(2.117999548\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.21 + 4.48i)T \)
good7 \( 1 + (-8.02 - 4.63i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-13.2 - 7.64i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.59 + 1.5i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 4.00T + 289T^{2} \)
19 \( 1 + 27.7T + 361T^{2} \)
23 \( 1 + (-18.9 - 32.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.7 - 22.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.3 + 24.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 34.3iT - 1.36e3T^{2} \)
41 \( 1 + (23.6 - 13.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (31.4 + 18.1i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (10.0 - 17.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 57.9T + 2.80e3T^{2} \)
59 \( 1 + (73.8 - 42.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.8 + 32.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (50.4 - 29.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 - 31.6iT - 5.32e3T^{2} \)
79 \( 1 + (19.5 - 33.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (32.9 - 57.0i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 85.2iT - 7.92e3T^{2} \)
97 \( 1 + (-140. - 80.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−9.023553530037121101192092650003, −8.635990265328636053711081423973, −7.88648048554508180057357125212, −6.98339815562258377036302397876, −5.98405124276459487822682601157, −4.96933596133966003468218800148, −4.51012018644183373160854057915, −3.47094296037866424332872490442, −1.91732524630664894685751693561, −1.19844384778697969055961385305, 0.64078660466263044324451612811, 1.90064006757911678641190169037, 3.14726182401019327427723308907, 4.11864078650951837812665975251, 4.68110889940962707420840682552, 6.11935416370927728998766525660, 6.70060933017842432245242895271, 7.44466243213782873289246469829, 8.505230512293229563202425940162, 8.708590028392562323700403260362

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