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Properties

Label 2-675-45.23-c1-0-8
Degree $2$
Conductor $675$
Sign $0.970 - 0.241i$
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  + (0.0499 − 0.186i)2-s + (1.69 + 0.981i)4-s + (2.35 + 0.632i)7-s + (0.540 − 0.540i)8-s + (2.14 − 1.23i)11-s + (−1.57 + 0.422i)13-s + (0.235 − 0.407i)14-s + (1.88 + 3.27i)16-s + (−0.403 − 0.403i)17-s − 4.28i·19-s + (−0.123 − 0.461i)22-s + (1.82 + 6.82i)23-s + 0.314i·26-s + (3.38 + 3.38i)28-s + (−3.20 − 5.55i)29-s + ⋯
L(s)  = 1  + (0.0352 − 0.131i)2-s + (0.849 + 0.490i)4-s + (0.891 + 0.238i)7-s + (0.191 − 0.191i)8-s + (0.646 − 0.373i)11-s + (−0.436 + 0.117i)13-s + (0.0629 − 0.108i)14-s + (0.472 + 0.818i)16-s + (−0.0979 − 0.0979i)17-s − 0.983i·19-s + (−0.0263 − 0.0982i)22-s + (0.381 + 1.42i)23-s + 0.0616i·26-s + (0.640 + 0.640i)28-s + (−0.595 − 1.03i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06864 + 0.253536i\)
\(L(\frac12)\) \(\approx\) \(2.06864 + 0.253536i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0499 + 0.186i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (-2.35 - 0.632i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.14 + 1.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.57 - 0.422i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.403 + 0.403i)T + 17iT^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 + (-1.82 - 6.82i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.20 + 5.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.97 - 3.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 + (-6.52 - 3.76i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.32 - 4.95i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.780 + 2.91i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.12 + 6.12i)T - 53iT^{2} \)
59 \( 1 + (2.27 - 3.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.235 + 0.408i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.443 - 1.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.50iT - 71T^{2} \)
73 \( 1 + (6.88 + 6.88i)T + 73iT^{2} \)
79 \( 1 + (-6.50 + 3.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.6 + 2.85i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.90T + 89T^{2} \)
97 \( 1 + (1.41 + 0.379i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.83781365465793328927561250112, −9.629916960336899558304489337311, −8.771626068698913073733774447009, −7.79573706175035296135563100741, −7.14303868299745092810578264091, −6.13359157674987851078957920648, −5.04367056680488072678628708756, −3.87970262790739554608676960765, −2.71708949819116583842294733342, −1.55433127574895013178250401122, 1.36246924119658506915310848453, 2.45061268525804421988124188256, 4.01990221643785106728523748110, 5.07914970207357331832046604313, 6.01325869130161664839978559819, 7.00189244305514016761869661355, 7.65736934674873341257992780318, 8.679387247170815466478592870822, 9.740504841550278145399776144126, 10.63041909315773869538876711378

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