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Properties

Label 2-1620-9.2-c2-0-31
Degree $2$
Conductor $1620$
Sign $-0.939 + 0.342i$
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  + (1.93 − 1.11i)5-s + (6.74 − 11.6i)7-s + (−15.2 − 8.82i)11-s + (3.74 + 6.48i)13-s − 16.9i·17-s − 10.9·19-s + (18.9 − 10.9i)23-s + (2.5 − 4.33i)25-s + (−41.0 − 23.6i)29-s + (8.48 + 14.6i)31-s − 30.1i·35-s − 5.53·37-s + (−57.4 + 33.1i)41-s + (−19.4 + 33.7i)43-s + (28.2 + 16.2i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.223i)5-s + (0.963 − 1.66i)7-s + (−1.39 − 0.802i)11-s + (0.287 + 0.498i)13-s − 0.998i·17-s − 0.577·19-s + (0.824 − 0.476i)23-s + (0.100 − 0.173i)25-s + (−1.41 − 0.816i)29-s + (0.273 + 0.474i)31-s − 0.861i·35-s − 0.149·37-s + (−1.40 + 0.809i)41-s + (−0.453 + 0.784i)43-s + (0.600 + 0.346i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.374421973\)
\(L(\frac12)\) \(\approx\) \(1.374421973\)
\(L(2)\) 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 \( 1 + (-1.93 + 1.11i)T \)
good7 \( 1 + (-6.74 + 11.6i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.2 + 8.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.74 - 6.48i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 16.9iT - 289T^{2} \)
19 \( 1 + 10.9T + 361T^{2} \)
23 \( 1 + (-18.9 + 10.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (41.0 + 23.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.48 - 14.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 5.53T + 1.36e3T^{2} \)
41 \( 1 + (57.4 - 33.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (19.4 - 33.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-28.2 - 16.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 11.2iT - 2.80e3T^{2} \)
59 \( 1 + (-27.6 + 15.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (23.4 - 40.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38 - 65.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 77.7iT - 5.04e3T^{2} \)
73 \( 1 - 94.9T + 5.32e3T^{2} \)
79 \( 1 + (-3.46 + 5.99i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (53.8 + 31.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (-62.4 + 108. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.690367826631326153171262697683, −8.052660230910477516979098856486, −7.32911325350770594567493409264, −6.56255996462237050667498514630, −5.35276334237062403725411115184, −4.76771016102924822009430100320, −3.85671673968320552188671696668, −2.67589824507794425130572866760, −1.40840800049109976359730899775, −0.35051889138418589714406698141, 1.81721353566779621917536735162, 2.31502754981315103068493336353, 3.48592237570551463113399505813, 5.00525347366469940126555526866, 5.32894273157849522077763458872, 6.13058441404719314984038512019, 7.28763076786375536340478920812, 8.105136626751905733190483985134, 8.687205355240178388672670163676, 9.456040058428473598260214755617

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