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Properties

Label 2-675-9.7-c1-0-7
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: $\chi_{675} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.921103 + 0.586807i\)
\(L(\frac12)\) \(\approx\) \(0.921103 + 0.586807i\)
\(L(\frac{3}{2})\) 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

−10.47275540001734202040571177524, −9.378686346450448002738193788218, −8.844317876905587566540507866837, −7.998307451086698871057874755440, −7.22489266843282878903724536965, −6.28425932539660894752703664990, −5.70436893726477482120889449979, −4.29670226883442904491842142145, −3.05808410808527275324793207420, −0.977560419106518621726127019490, 1.11625810693004984083658261433, 2.28709588926291308269007573554, 3.38379453080188592334922313996, 4.58227582255976799345221496739, 5.82495583358172831839987622950, 6.93614685958454407483425597941, 8.025788166271398255878953188048, 8.940235497955483844248240899343, 9.613198657547099261052347647904, 10.31681646536398190972041991312

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