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Properties

Label 2-675-9.4-c1-0-5
Degree $2$
Conductor $675$
Sign $-0.422 - 0.906i$
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  + (0.965 + 1.67i)2-s + (−0.866 + 1.50i)4-s + (−0.448 − 0.776i)7-s + 0.517·8-s + (2.36 + 4.09i)11-s + (−1.22 + 2.12i)13-s + (0.866 − 1.50i)14-s + (2.23 + 3.86i)16-s − 0.378·17-s + 2.73·19-s + (−4.57 + 7.91i)22-s + (−0.965 + 1.67i)23-s − 4.73·26-s + 1.55·28-s + (3.23 + 5.59i)29-s + ⋯
L(s)  = 1  + (0.683 + 1.18i)2-s + (−0.433 + 0.750i)4-s + (−0.169 − 0.293i)7-s + 0.183·8-s + (0.713 + 1.23i)11-s + (−0.339 + 0.588i)13-s + (0.231 − 0.400i)14-s + (0.558 + 0.966i)16-s − 0.0919·17-s + 0.626·19-s + (−0.974 + 1.68i)22-s + (−0.201 + 0.348i)23-s − 0.928·26-s + 0.293·28-s + (0.600 + 1.03i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.422 - 0.906i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19679 + 1.87859i\)
\(L(\frac12)\) \(\approx\) \(1.19679 + 1.87859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 - 1.67i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.448 + 0.776i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.22 - 2.12i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.378T + 17T^{2} \)
19 \( 1 - 2.73T + 19T^{2} \)
23 \( 1 + (0.965 - 1.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.36 + 2.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + (-2.13 + 3.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.57 + 7.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.19 - 3.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.86T + 53T^{2} \)
59 \( 1 + (-1.26 + 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.33 + 9.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.56 + 4.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.80T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + (0.267 + 0.464i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.20 + 9.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.39T + 89T^{2} \)
97 \( 1 + (-5.13 - 8.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68200119395794169365166792003, −9.809215947615671985676477114899, −8.945782859270132816119836679924, −7.76122910580192100408346460275, −7.02620693108412425637031982750, −6.53001757319412689187359023563, −5.32433701520392226666786256257, −4.56328956952657579550627862659, −3.64039194277109783683069256531, −1.78709758585820404782830364033, 1.07305219737098340952357839981, 2.62059700382854803220302078075, 3.37758279118044611017940782397, 4.41667318973970983146238965452, 5.47835885832895050020856918654, 6.41081928565215085778617566919, 7.70196107620916238376818412376, 8.635676920203475041599337993574, 9.667313453846490437495794586126, 10.42411402827947648336069733602

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