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Properties

Label 2-2700-45.14-c2-0-26
Degree $2$
Conductor $2700$
Sign $-0.549 + 0.835i$
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  + (−7.02 + 4.05i)7-s + (17.6 − 10.1i)11-s + (5.29 + 3.05i)13-s − 17.9·17-s − 9.11·19-s + (−16.7 + 29.0i)23-s + (14.4 − 8.31i)29-s + (11.1 − 19.3i)31-s − 50.4i·37-s + (−29.9 − 17.3i)41-s + (−19.9 + 11.5i)43-s + (19.1 + 33.1i)47-s + (8.44 − 14.6i)49-s + 19.0·53-s + (−2.96 − 1.71i)59-s + ⋯
L(s)  = 1  + (−1.00 + 0.579i)7-s + (1.60 − 0.924i)11-s + (0.407 + 0.235i)13-s − 1.05·17-s − 0.479·19-s + (−0.729 + 1.26i)23-s + (0.496 − 0.286i)29-s + (0.360 − 0.624i)31-s − 1.36i·37-s + (−0.730 − 0.421i)41-s + (−0.463 + 0.267i)43-s + (0.407 + 0.705i)47-s + (0.172 − 0.298i)49-s + 0.358·53-s + (−0.0502 − 0.0290i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.549 + 0.835i$
Analytic conductor: \(73.5696\)
Root analytic conductor: \(8.57727\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1),\ -0.549 + 0.835i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7393922566\)
\(L(\frac12)\) \(\approx\) \(0.7393922566\)
\(L(2)\) 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 \( 1 + (7.02 - 4.05i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-17.6 + 10.1i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.29 - 3.05i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 17.9T + 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (16.7 - 29.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-14.4 + 8.31i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.1 + 19.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 50.4iT - 1.36e3T^{2} \)
41 \( 1 + (29.9 + 17.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (19.9 - 11.5i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-19.1 - 33.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 19.0T + 2.80e3T^{2} \)
59 \( 1 + (2.96 + 1.71i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 - 40.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.45 - 3.14i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 35.9iT - 5.04e3T^{2} \)
73 \( 1 + 47.3iT - 5.32e3T^{2} \)
79 \( 1 + (42.2 + 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (19.1 + 33.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (-69.9 + 40.3i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−8.698591775173130017090754220203, −7.64798470933708973955197361358, −6.49230328619741803707223393080, −6.34693307092815360395379364279, −5.51193251140920596334439568190, −4.14735440262232254108591100753, −3.69063266375644309790202447304, −2.63686160634355571847129454199, −1.52256936171998889311170268169, −0.18359636844966759318812659645, 1.11113964620500430683425538324, 2.23969081869696474725950270274, 3.41132478397818561941836392343, 4.14972599024493057855281325610, 4.79019161669907923711274509196, 6.21644944039622329359937112931, 6.66210964716788287588260347419, 7.05726670700973477927751197832, 8.439561286257206433522194486735, 8.769996612406120871133460386975

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