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Properties

Label 2-30e2-15.2-c3-0-14
Degree $2$
Conductor $900$
Sign $0.0387 + 0.999i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
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  + (17.1 + 17.1i)7-s − 68.0i·11-s + (−33.1 + 33.1i)13-s + (15.5 − 15.5i)17-s − 40.1i·19-s + (−52.1 − 52.1i)23-s + 97.9·29-s − 206.·31-s + (−238. − 238. i)37-s + 43.1i·41-s + (255. − 255. i)43-s + (134. − 134. i)47-s + 248. i·49-s + (458. + 458. i)53-s − 748.·59-s + ⋯
L(s)  = 1  + (0.928 + 0.928i)7-s − 1.86i·11-s + (−0.707 + 0.707i)13-s + (0.222 − 0.222i)17-s − 0.484i·19-s + (−0.473 − 0.473i)23-s + 0.627·29-s − 1.19·31-s + (−1.05 − 1.05i)37-s + 0.164i·41-s + (0.905 − 0.905i)43-s + (0.418 − 0.418i)47-s + 0.724i·49-s + (1.18 + 1.18i)53-s − 1.65·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0387 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0387 + 0.999i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ 0.0387 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.579885184\)
\(L(\frac12)\) \(\approx\) \(1.579885184\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good7 \( 1 + (-17.1 - 17.1i)T + 343iT^{2} \)
11 \( 1 + 68.0iT - 1.33e3T^{2} \)
13 \( 1 + (33.1 - 33.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-15.5 + 15.5i)T - 4.91e3iT^{2} \)
19 \( 1 + 40.1iT - 6.85e3T^{2} \)
23 \( 1 + (52.1 + 52.1i)T + 1.21e4iT^{2} \)
29 \( 1 - 97.9T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + (238. + 238. i)T + 5.06e4iT^{2} \)
41 \( 1 - 43.1iT - 6.89e4T^{2} \)
43 \( 1 + (-255. + 255. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-134. + 134. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-458. - 458. i)T + 1.48e5iT^{2} \)
59 \( 1 + 748.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (-133. - 133. i)T + 3.00e5iT^{2} \)
71 \( 1 + 899. iT - 3.57e5T^{2} \)
73 \( 1 + (-26.5 + 26.5i)T - 3.89e5iT^{2} \)
79 \( 1 - 112. iT - 4.93e5T^{2} \)
83 \( 1 + (-644. - 644. i)T + 5.71e5iT^{2} \)
89 \( 1 - 554.T + 7.04e5T^{2} \)
97 \( 1 + (1.24e3 + 1.24e3i)T + 9.12e5iT^{2} \)
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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

−9.181052373098555176278641534776, −8.803609791255592402932781870812, −7.971171825209446446217536901543, −6.98375315947063838075564387839, −5.82861457898438008370640878104, −5.29532948906248526461223308739, −4.14618044877286520071670653396, −2.91280360849599240657503342069, −1.90895040515582567385158641226, −0.41393282574874424821359457365, 1.25407469252762938236519844885, 2.23694854477131762776104267528, 3.74961300394325470413467176473, 4.64512595681195551377016405376, 5.32696728874822102606253167486, 6.70570103952898766347201901030, 7.59195115651337951199075938467, 7.88382619098191569914288305039, 9.206735703692907144312837204553, 10.17012665787037479275042238483

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