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Properties

Label 2-675-45.4-c1-0-8
Degree $2$
Conductor $675$
Sign $0.993 - 0.114i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (0.866 − 0.5i)2-s + (−0.500 + 0.866i)4-s + (2.59 − 1.5i)7-s + 3i·8-s + (−1 − 1.73i)11-s + (1.73 + i)13-s + (1.5 − 2.59i)14-s + (0.500 + 0.866i)16-s + 4i·17-s + 8·19-s + (−1.73 − 0.999i)22-s + (2.59 + 1.5i)23-s + 1.99·26-s + 3i·28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.250 + 0.433i)4-s + (0.981 − 0.566i)7-s + 1.06i·8-s + (−0.301 − 0.522i)11-s + (0.480 + 0.277i)13-s + (0.400 − 0.694i)14-s + (0.125 + 0.216i)16-s + 0.970i·17-s + 1.83·19-s + (−0.369 − 0.213i)22-s + (0.541 + 0.312i)23-s + 0.392·26-s + 0.566i·28-s + (0.0928 + 0.160i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14741 + 0.123162i\)
\(L(\frac12)\) \(\approx\) \(2.14741 + 0.123162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + (-7 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (3 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (1.73 - i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.90642698527740960415877920159, −9.713116724960405718340243398995, −8.572068195593347711592871001881, −8.019724471807744644162704460656, −7.12338386369203983760094452322, −5.68368825407470785520033138161, −4.91424739737909736192938338204, −3.91994113874144726662661087229, −3.05509898669358670064665512742, −1.47004028892139664765682362741, 1.20513933024391575085177907668, 2.84034043073368269531378945440, 4.25138102848267378950118637080, 5.21228152956657950829616000738, 5.59384290486228412745919484861, 6.92292644948889585028588805074, 7.69455030349641775089774965689, 8.823099313681104013352343971600, 9.586403667212809748677897649958, 10.45420552087374898278983221479

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