↑

Properties

Label 2-675-9.7-c1-0-1
Degree $2$
Conductor $675$
Sign $-0.906 - 0.422i$
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  + (−0.258 + 0.448i)2-s + (0.866 + 1.5i)4-s + (−1.67 + 2.89i)7-s − 1.93·8-s + (0.633 − 1.09i)11-s + (1.22 + 2.12i)13-s + (−0.866 − 1.5i)14-s + (−1.23 + 2.13i)16-s − 5.27·17-s − 0.732·19-s + (0.328 + 0.568i)22-s + (0.258 + 0.448i)23-s − 1.26·26-s − 5.79·28-s + (−0.232 + 0.401i)29-s + ⋯
L(s)  = 1  + (−0.183 + 0.316i)2-s + (0.433 + 0.750i)4-s + (−0.632 + 1.09i)7-s − 0.683·8-s + (0.191 − 0.331i)11-s + (0.339 + 0.588i)13-s + (−0.231 − 0.400i)14-s + (−0.308 + 0.533i)16-s − 1.28·17-s − 0.167·19-s + (0.0699 + 0.121i)22-s + (0.0539 + 0.0934i)23-s − 0.248·26-s − 1.09·28-s + (−0.0430 + 0.0746i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.906 - 0.422i$
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: \((2,\ 675,\ (\ :1/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216833 + 0.978072i\)
\(L(\frac12)\) \(\approx\) \(0.216833 + 0.978072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.448i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.67 - 2.89i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.27T + 17T^{2} \)
19 \( 1 + 0.732T + 19T^{2} \)
23 \( 1 + (-0.258 - 0.448i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.366 + 0.633i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + (-3.86 - 6.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.328 + 0.568i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.48 - 2.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 + (-4.73 - 8.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.33 + 5.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.79 - 6.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 + (3.73 - 6.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.98 - 6.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (-7.58 + 13.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.16255883556591949424398299229, −9.783277492540832975407673677176, −8.815972678069108204681614557059, −8.562645169051612524244016222993, −7.25784219809601635748244681984, −6.47580750921813254304037702746, −5.80353683853823992321598784781, −4.31531077964427391843823362772, −3.16068261086876051483796408053, −2.18320602064042363591302582419, 0.53051108341858227404298957611, 2.02031537197286078017847787873, 3.36274052049625360057047830723, 4.48830709683416281185214010457, 5.72597209231435576499471838774, 6.67920650291524285363016434506, 7.23190626116314871149129041331, 8.581696959469217447941594105781, 9.473387128028999580163858651684, 10.30393136080488373453234458115

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