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Properties

Label 2-675-5.4-c3-0-12
Degree $2$
Conductor $675$
Sign $-0.894 + 0.447i$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
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  + 2.58i·2-s + 1.31·4-s + 22.8i·7-s + 24.0i·8-s + 11.0·11-s − 11.6i·13-s − 59.1·14-s − 51.7·16-s − 10.0i·17-s − 117.·19-s + 28.6i·22-s + 172. i·23-s + 30.0·26-s + 30.1i·28-s − 178.·29-s + ⋯
L(s)  = 1  + 0.913i·2-s + 0.164·4-s + 1.23i·7-s + 1.06i·8-s + 0.303·11-s − 0.248i·13-s − 1.12·14-s − 0.808·16-s − 0.143i·17-s − 1.42·19-s + 0.277i·22-s + 1.56i·23-s + 0.226·26-s + 0.203i·28-s − 1.14·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.453256154\)
\(L(\frac12)\) \(\approx\) \(1.453256154\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 2.58iT - 8T^{2} \)
7 \( 1 - 22.8iT - 343T^{2} \)
11 \( 1 - 11.0T + 1.33e3T^{2} \)
13 \( 1 + 11.6iT - 2.19e3T^{2} \)
17 \( 1 + 10.0iT - 4.91e3T^{2} \)
19 \( 1 + 117.T + 6.85e3T^{2} \)
23 \( 1 - 172. iT - 1.21e4T^{2} \)
29 \( 1 + 178.T + 2.43e4T^{2} \)
31 \( 1 - 140.T + 2.97e4T^{2} \)
37 \( 1 + 250. iT - 5.06e4T^{2} \)
41 \( 1 - 361.T + 6.89e4T^{2} \)
43 \( 1 + 360. iT - 7.95e4T^{2} \)
47 \( 1 - 600. iT - 1.03e5T^{2} \)
53 \( 1 - 201. iT - 1.48e5T^{2} \)
59 \( 1 + 415.T + 2.05e5T^{2} \)
61 \( 1 + 54.6T + 2.26e5T^{2} \)
67 \( 1 - 531. iT - 3.00e5T^{2} \)
71 \( 1 + 933.T + 3.57e5T^{2} \)
73 \( 1 + 560. iT - 3.89e5T^{2} \)
79 \( 1 + 810.T + 4.93e5T^{2} \)
83 \( 1 - 538. iT - 5.71e5T^{2} \)
89 \( 1 + 686.T + 7.04e5T^{2} \)
97 \( 1 + 714. iT - 9.12e5T^{2} \)
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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

−10.71535613216399384800061857413, −9.367989825220857471792165390705, −8.799488947953234537530906703255, −7.85950776342257557881313205477, −7.10448505215200947026377489049, −5.87852283882291713772852386030, −5.73262374348171215755014670654, −4.35381101742943076546122528262, −2.83573671523656158284193952664, −1.83416561076099519951102046025, 0.37847079169619703003743954056, 1.54704079918297502198978882231, 2.69941418085353006708999976196, 3.92813040836608617422081410410, 4.49567327556628627583179645057, 6.27979001321018870947768183225, 6.82875560223043451427566158645, 7.85286918587885619655587448056, 8.916302384514873373216237367001, 10.00085151633201173230813791026

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