Properties

Label 2-1575-35.34-c0-0-0
Degree $2$
Conductor $1575$
Sign $-0.447 + 0.894i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·2-s − 1.99·4-s + i·7-s − 1.73i·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99i·22-s + 1.73i·23-s − 1.99i·28-s − 1.73·29-s i·37-s + i·43-s + 3.46·44-s − 2.99·46-s + ⋯
L(s)  = 1  + 1.73i·2-s − 1.99·4-s + i·7-s − 1.73i·8-s − 1.73·11-s − 1.73·14-s + 0.999·16-s − 2.99i·22-s + 1.73i·23-s − 1.99i·28-s − 1.73·29-s i·37-s + i·43-s + 3.46·44-s − 2.99·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5690996660\)
\(L(\frac12)\) \(\approx\) \(0.5690996660\)
\(L(1)\) 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 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 - 1.73iT - T^{2} \)
11 \( 1 + 1.73T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.629071067753199038410813677067, −9.221814849739519169053488456851, −8.215562944474450010808414124640, −7.75774232810408546765934207659, −7.08109394419124985682647012776, −5.90907820380374481390675623890, −5.51439957514368149140919671595, −4.89914503614370000537205674676, −3.57758529889259728472204541316, −2.26010119534454783370060862338, 0.42846022477423638282803820462, 1.93883202684769553061337964284, 2.83376678204894359963705008112, 3.74285165313105776761812360269, 4.61270815220857831736936482865, 5.35172684134976090536248094010, 6.73575625616169407233442560081, 7.77289448354688003168419202219, 8.437682347195741148827930487464, 9.476046377959525770955781843709

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