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Properties

Label 2-2700-45.29-c2-0-28
Degree $2$
Conductor $2700$
Sign $0.0969 + 0.995i$
Analytic cond. $73.5696$
Root an. cond. $8.57727$
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  + (1.02 + 0.594i)7-s + (−3.63 − 2.09i)11-s + (12.3 − 7.15i)13-s + 18.7·17-s − 33.8·19-s + (−6.74 − 11.6i)23-s + (30.6 + 17.6i)29-s + (4.97 + 8.62i)31-s − 19.3i·37-s + (55.9 − 32.3i)41-s + (−35.9 − 20.7i)43-s + (−33.6 + 58.2i)47-s + (−23.7 − 41.2i)49-s − 30.0·53-s + (3.66 − 2.11i)59-s + ⋯
L(s)  = 1  + (0.147 + 0.0849i)7-s + (−0.330 − 0.190i)11-s + (0.953 − 0.550i)13-s + 1.10·17-s − 1.78·19-s + (−0.293 − 0.508i)23-s + (1.05 + 0.609i)29-s + (0.160 + 0.278i)31-s − 0.521i·37-s + (1.36 − 0.788i)41-s + (−0.836 − 0.483i)43-s + (−0.715 + 1.23i)47-s + (−0.485 − 0.841i)49-s − 0.567·53-s + (0.0621 − 0.0358i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0969 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0969 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.0969 + 0.995i$
Analytic conductor: \(73.5696\)
Root analytic conductor: \(8.57727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1),\ 0.0969 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.650499317\)
\(L(\frac12)\) \(\approx\) \(1.650499317\)
\(L(2)\) 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 \( 1 \)
good7 \( 1 + (-1.02 - 0.594i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.63 + 2.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-12.3 + 7.15i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 18.7T + 289T^{2} \)
19 \( 1 + 33.8T + 361T^{2} \)
23 \( 1 + (6.74 + 11.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-30.6 - 17.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4.97 - 8.62i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 19.3iT - 1.36e3T^{2} \)
41 \( 1 + (-55.9 + 32.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (35.9 + 20.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (33.6 - 58.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 30.0T + 2.80e3T^{2} \)
59 \( 1 + (-3.66 + 2.11i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-43.8 + 75.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (31.0 - 17.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 24.5iT - 5.04e3T^{2} \)
73 \( 1 + 11.9iT - 5.32e3T^{2} \)
79 \( 1 + (19.0 - 33.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (12.0 - 20.8i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 44.4iT - 7.92e3T^{2} \)
97 \( 1 + (55.3 + 31.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−8.284035126208674446098297996535, −8.047476620193257634045416946752, −6.85437245576742101368059540280, −6.17660142799587923746073857128, −5.44636294442619915410271339824, −4.52991446778488854053840149415, −3.62774125888353901539324021422, −2.74821128483084143390222464140, −1.61846675772300319417789653661, −0.42021885331148975092288878224, 1.07364724046793029542450508547, 2.10157714688038608399050759158, 3.18709941044185893530797352141, 4.14967756025427965947446109062, 4.80175659051817637243906716898, 5.97330053827247024729002955865, 6.37529222695671721222386083715, 7.39225996191121801171396209590, 8.217341353971579247410917445881, 8.623110579551229838415284830221

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