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Properties

Label 2-1620-9.7-c3-0-46
Degree $2$
Conductor $1620$
Sign $-0.939 + 0.342i$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
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  + (−2.5 − 4.33i)5-s + (8 − 13.8i)7-s + (30 − 51.9i)11-s + (−43 − 74.4i)13-s + 18·17-s + 44·19-s + (−24 − 41.5i)23-s + (−12.5 + 21.6i)25-s + (93 − 161. i)29-s + (−88 − 152. i)31-s − 80·35-s + 254·37-s + (−93 − 161. i)41-s + (50 − 86.6i)43-s + (−84 + 145. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.431 − 0.748i)7-s + (0.822 − 1.42i)11-s + (−0.917 − 1.58i)13-s + 0.256·17-s + 0.531·19-s + (−0.217 − 0.376i)23-s + (−0.100 + 0.173i)25-s + (0.595 − 1.03i)29-s + (−0.509 − 0.883i)31-s − 0.386·35-s + 1.12·37-s + (−0.354 − 0.613i)41-s + (0.177 − 0.307i)43-s + (−0.260 + 0.451i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.853609839\)
\(L(\frac12)\) \(\approx\) \(1.853609839\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.5 + 4.33i)T \)
good7 \( 1 + (-8 + 13.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-30 + 51.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (43 + 74.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 18T + 4.91e3T^{2} \)
19 \( 1 - 44T + 6.85e3T^{2} \)
23 \( 1 + (24 + 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-93 + 161. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (88 + 152. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 254T + 5.06e4T^{2} \)
41 \( 1 + (93 + 161. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-50 + 86.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (84 - 145. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 498T + 1.48e5T^{2} \)
59 \( 1 + (-126 - 218. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-29 + 50.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-518 - 897. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 168T + 3.57e5T^{2} \)
73 \( 1 - 506T + 3.89e5T^{2} \)
79 \( 1 + (136 - 235. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (474 - 820. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + (-383 + 663. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−8.467532787410794763303008560621, −7.957109355746499009937767322974, −7.27620411286742833609475358593, −6.09592068770082499213740821250, −5.45064803218611898026405339361, −4.43281296157921814799798921722, −3.60725250117881710890548076677, −2.63686675039860472669532353469, −1.02762245560331568968348159356, −0.46210142232394493575342923117, 1.53297698433919857716908294045, 2.22207379502385187622158544115, 3.44839664020609712214412622713, 4.56271217361229239488727012135, 5.04330283727452975346653873661, 6.36714415505822392502056569610, 6.98955959493908023283450920416, 7.64555179667562509002798107015, 8.699933060971299745678436256583, 9.497586205429292928245076863332

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