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Properties

Label 2-2700-15.2-c1-0-0
Degree $2$
Conductor $2700$
Sign $-0.973 + 0.229i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.79 + 1.79i)7-s + 0.646i·11-s + (−3.79 + 3.79i)13-s + (−4.96 + 4.96i)17-s − 6.58i·19-s + (−1.87 − 1.87i)23-s − 7.99·29-s − 0.582·31-s + (−3 − 3i)37-s − 4.25i·41-s + (0.791 − 0.791i)43-s + (1.93 − 1.93i)47-s − 0.582i·49-s + (−5.47 − 5.47i)53-s − 12.8·59-s + ⋯
L(s)  = 1  + (0.677 + 0.677i)7-s + 0.194i·11-s + (−1.05 + 1.05i)13-s + (−1.20 + 1.20i)17-s − 1.51i·19-s + (−0.390 − 0.390i)23-s − 1.48·29-s − 0.104·31-s + (−0.493 − 0.493i)37-s − 0.664i·41-s + (0.120 − 0.120i)43-s + (0.282 − 0.282i)47-s − 0.0832i·49-s + (−0.752 − 0.752i)53-s − 1.67·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1558457944\)
\(L(\frac12)\) \(\approx\) \(0.1558457944\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 1.79i)T + 7iT^{2} \)
11 \( 1 - 0.646iT - 11T^{2} \)
13 \( 1 + (3.79 - 3.79i)T - 13iT^{2} \)
17 \( 1 + (4.96 - 4.96i)T - 17iT^{2} \)
19 \( 1 + 6.58iT - 19T^{2} \)
23 \( 1 + (1.87 + 1.87i)T + 23iT^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 + 0.582T + 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + 4.25iT - 41T^{2} \)
43 \( 1 + (-0.791 + 0.791i)T - 43iT^{2} \)
47 \( 1 + (-1.93 + 1.93i)T - 47iT^{2} \)
53 \( 1 + (5.47 + 5.47i)T + 53iT^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 6.16T + 61T^{2} \)
67 \( 1 + (-7 - 7i)T + 67iT^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + (-1.20 + 1.20i)T - 73iT^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 + (-1.22 - 1.22i)T + 83iT^{2} \)
89 \( 1 - 9.42T + 89T^{2} \)
97 \( 1 + (7.58 + 7.58i)T + 97iT^{2} \)
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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

−9.031999984659882729573135360441, −8.723752147407684954269146971091, −7.69309804994885997189944499335, −6.96750765671747115776475706212, −6.26776376308000307711748440564, −5.21835777733531426113894089283, −4.62931197647700243084736442512, −3.78715873206343732224675429082, −2.28722100813272669889005517303, −1.95844297889775395239319350633, 0.04659057927145982793336365855, 1.47608738147509487381440740096, 2.58242177158613518127357205166, 3.61402540954691726899934425292, 4.54031532898446085563662013139, 5.23702046573544124443361755333, 6.05667952300360790108689412784, 7.12942874414371913195654955227, 7.68920134489268855796335723243, 8.214438745507657297060282399132

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