↑

Properties

Label 2-675-15.8-c1-0-12
Degree $2$
Conductor $675$
Sign $-0.973 - 0.229i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 1.79i)2-s + 4.44i·4-s + (−2.22 + 2.22i)7-s + (4.39 − 4.39i)8-s − 0.807i·11-s + (0.775 + 0.775i)13-s + 7.99·14-s − 6.89·16-s + (−0.807 − 0.807i)17-s − 3i·19-s + (−1.44 + 1.44i)22-s + (1.79 − 1.79i)23-s − 2.78i·26-s + (−9.89 − 9.89i)28-s − 6.37·29-s + ⋯
L(s)  = 1  + (−1.26 − 1.26i)2-s + 2.22i·4-s + (−0.840 + 0.840i)7-s + (1.55 − 1.55i)8-s − 0.243i·11-s + (0.215 + 0.215i)13-s + 2.13·14-s − 1.72·16-s + (−0.195 − 0.195i)17-s − 0.688i·19-s + (−0.309 + 0.309i)22-s + (0.374 − 0.374i)23-s − 0.546i·26-s + (−1.87 − 1.87i)28-s − 1.18·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0290491 + 0.249490i\)
\(L(\frac12)\) \(\approx\) \(0.0290491 + 0.249490i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.79 + 1.79i)T + 2iT^{2} \)
7 \( 1 + (2.22 - 2.22i)T - 7iT^{2} \)
11 \( 1 + 0.807iT - 11T^{2} \)
13 \( 1 + (-0.775 - 0.775i)T + 13iT^{2} \)
17 \( 1 + (0.807 + 0.807i)T + 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (-1.79 + 1.79i)T - 23iT^{2} \)
29 \( 1 + 6.37T + 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 + (-5.67 + 5.67i)T - 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (3.44 + 3.44i)T + 43iT^{2} \)
47 \( 1 + (7.18 + 7.18i)T + 47iT^{2} \)
53 \( 1 + (7.00 - 7.00i)T - 53iT^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-6.22 + 6.22i)T - 67iT^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-2.67 - 2.67i)T + 73iT^{2} \)
79 \( 1 - 0.550iT - 79T^{2} \)
83 \( 1 + (2.60 - 2.60i)T - 83iT^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (-2.12 + 2.12i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.946571174265909301068314167813, −9.087777812801270320043526140712, −8.921159163708330871831191389581, −7.72172064979832598645248212208, −6.76048375976678980636142265498, −5.50975669702119956402628874143, −3.87937916309513500933153237229, −2.92175924988487789928649943215, −1.96465373250865687154178706526, −0.21055746666800157000662563466, 1.39571908671945850645024373831, 3.46626303997742316842837949032, 4.90405816707781816766505321920, 6.11483878937391420257496382642, 6.63104270693024515527057119545, 7.62304755579983699801656385500, 8.108660899597596902452544710701, 9.371952169025554778693107779173, 9.694878828986412047779289606237, 10.57176077998088580637032689924

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