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Properties

Label 2-1620-9.7-c3-0-7
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 + (−16 + 27.7i)7-s + (−18 + 31.1i)11-s + (5 + 8.66i)13-s − 78·17-s + 140·19-s + (96 + 166. i)23-s + (−12.5 + 21.6i)25-s + (−3 + 5.19i)29-s + (8 + 13.8i)31-s + 160·35-s − 34·37-s + (195 + 337. i)41-s + (26 − 45.0i)43-s + (−204 + 353. i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (−0.863 + 1.49i)7-s + (−0.493 + 0.854i)11-s + (0.106 + 0.184i)13-s − 1.11·17-s + 1.69·19-s + (0.870 + 1.50i)23-s + (−0.100 + 0.173i)25-s + (−0.0192 + 0.0332i)29-s + (0.0463 + 0.0802i)31-s + 0.772·35-s − 0.151·37-s + (0.742 + 1.28i)41-s + (0.0922 − 0.159i)43-s + (−0.633 + 1.09i)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

\(L(2)\) \(\approx\) \(0.6558132106\)
\(L(\frac12)\) \(\approx\) \(0.6558132106\)
\(L(\frac{5}{2})\) 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 \( 1 + (16 - 27.7i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (18 - 31.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-5 - 8.66i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 78T + 4.91e3T^{2} \)
19 \( 1 - 140T + 6.85e3T^{2} \)
23 \( 1 + (-96 - 166. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-8 - 13.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 34T + 5.06e4T^{2} \)
41 \( 1 + (-195 - 337. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-26 + 45.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (204 - 353. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 114T + 1.48e5T^{2} \)
59 \( 1 + (258 + 446. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-29 + 50.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-446 - 772. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 120T + 3.57e5T^{2} \)
73 \( 1 + 646T + 3.89e5T^{2} \)
79 \( 1 + (-584 + 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-366 + 633. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.59e3T + 7.04e5T^{2} \)
97 \( 1 + (97 - 168. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−9.406387598840837422606943302934, −8.905214130167463235394567076259, −7.87076860642671868093241691932, −7.11313868457685131755454460130, −6.16464000749535693100044808661, −5.35526127206624336186457874705, −4.69223559190082001659282704677, −3.35255608716839868316821000213, −2.61576544502341411135940324774, −1.45812535257330963835367908214, 0.17421488488855393119858874445, 0.916182371466648722003163065475, 2.67356631562821885401949476438, 3.41899285004078726050448338984, 4.22749967939101336837958978426, 5.27907501843995660221541260089, 6.38292251657889686978263648065, 6.98515823862389530334674548674, 7.63425758583522817761418093833, 8.572304199300811164115246499573

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