Properties

Label 2-1620-9.5-c2-0-14
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 + (−2.74 − 4.75i)7-s + (7.94 − 4.58i)11-s + (−5.74 + 9.94i)13-s + 16.9i·17-s + 26.9·19-s + (−4.27 − 2.46i)23-s + (2.5 + 4.33i)25-s + (−17.7 + 10.2i)29-s + (−10.4 + 18.1i)31-s + 12.2i·35-s − 62.4·37-s + (35.4 + 20.4i)41-s + (−0.513 − 0.888i)43-s + (74.6 − 43.1i)47-s + ⋯
L(s)  = 1  + (−0.387 − 0.223i)5-s + (−0.391 − 0.678i)7-s + (0.722 − 0.416i)11-s + (−0.441 + 0.765i)13-s + 0.998i·17-s + 1.41·19-s + (−0.185 − 0.107i)23-s + (0.100 + 0.173i)25-s + (−0.612 + 0.353i)29-s + (−0.338 + 0.585i)31-s + 0.350i·35-s − 1.68·37-s + (0.864 + 0.499i)41-s + (−0.0119 − 0.0206i)43-s + (1.58 − 0.917i)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: $\chi_{1620} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.693308552\)
\(L(\frac12)\) \(\approx\) \(1.693308552\)
\(L(2)\) 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 \( 1 + (2.74 + 4.75i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.94 + 4.58i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (5.74 - 9.94i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 - 26.9T + 361T^{2} \)
23 \( 1 + (4.27 + 2.46i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (17.7 - 10.2i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (10.4 - 18.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 62.4T + 1.36e3T^{2} \)
41 \( 1 + (-35.4 - 20.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (0.513 + 0.888i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-74.6 + 43.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 96.0iT - 2.80e3T^{2} \)
59 \( 1 + (-97.3 - 56.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-33.4 - 57.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-38 + 65.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 24.0iT - 5.04e3T^{2} \)
73 \( 1 + 18.9T + 5.32e3T^{2} \)
79 \( 1 + (53.4 + 92.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-39.1 + 22.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 115. iT - 7.92e3T^{2} \)
97 \( 1 + (-43.5 - 75.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.061687355235710128290788017261, −8.521657553733082458683283891330, −7.34152607601417114835362662411, −6.98776403530723190119035144392, −5.92711043987710945768349682364, −5.00627562773510862287258703660, −3.88693714812184564419958992234, −3.46820454802401170536619750071, −1.87890952806688220399252081192, −0.70381405719877988101267905707, 0.75902945607078682776009434895, 2.31633340594718057239299056996, 3.18987509467134113428894112702, 4.13362174230296697219546635515, 5.29446796313020055723145864157, 5.85785185436169778172587519251, 7.13442269269704812843328624978, 7.41262830534563124742075689488, 8.512544240766664351700999176486, 9.441895452467662710616756045927

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