↑

Properties

Label 2-675-15.14-c2-0-40
Degree $2$
Conductor $675$
Sign $-0.447 + 0.894i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·2-s + 1.00·4-s − 3i·7-s − 6.70·8-s − 4.47i·11-s − 14i·13-s − 6.70i·14-s − 19·16-s + 8.94·17-s − 19-s − 10.0i·22-s − 40.2·23-s − 31.3i·26-s − 3.00i·28-s − 44.7i·29-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.250·4-s − 0.428i·7-s − 0.838·8-s − 0.406i·11-s − 1.07i·13-s − 0.479i·14-s − 1.18·16-s + 0.526·17-s − 0.0526·19-s − 0.454i·22-s − 1.74·23-s − 1.20i·26-s − 0.107i·28-s − 1.54i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.815583250\)
\(L(\frac12)\) \(\approx\) \(1.815583250\)
\(L(2)\) 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 \( 1 - 2.23T + 4T^{2} \)
7 \( 1 + 3iT - 49T^{2} \)
11 \( 1 + 4.47iT - 121T^{2} \)
13 \( 1 + 14iT - 169T^{2} \)
17 \( 1 - 8.94T + 289T^{2} \)
19 \( 1 + T + 361T^{2} \)
23 \( 1 + 40.2T + 529T^{2} \)
29 \( 1 + 44.7iT - 841T^{2} \)
31 \( 1 + 21T + 961T^{2} \)
37 \( 1 + 53iT - 1.36e3T^{2} \)
41 \( 1 + 22.3iT - 1.68e3T^{2} \)
43 \( 1 + 23iT - 1.84e3T^{2} \)
47 \( 1 - 76.0T + 2.20e3T^{2} \)
53 \( 1 + 49.1T + 2.80e3T^{2} \)
59 \( 1 - 84.9iT - 3.48e3T^{2} \)
61 \( 1 + 71T + 3.72e3T^{2} \)
67 \( 1 - 6iT - 4.48e3T^{2} \)
71 \( 1 - 58.1iT - 5.04e3T^{2} \)
73 \( 1 - 7iT - 5.32e3T^{2} \)
79 \( 1 + 51T + 6.24e3T^{2} \)
83 \( 1 - 134.T + 6.88e3T^{2} \)
89 \( 1 - 120. iT - 7.92e3T^{2} \)
97 \( 1 + 143iT - 9.40e3T^{2} \)
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

−10.16255743748752636398234161286, −9.192575575189802691059522299976, −8.148258186772585958540898009910, −7.32087365318467071331573268257, −5.88691651202428615425875822941, −5.66085408961019734675751285266, −4.26949593145483256874962838247, −3.64827861353187630140503992795, −2.44139834843985781436361148486, −0.43033643952684534043090570363, 1.87576421691008904455670557710, 3.17624512096640311550926336907, 4.18044567464227724543259211654, 4.99708102541506452802442858831, 5.95289405468981075266034837568, 6.71862735146649881407029870043, 7.901165644685040664955958843601, 8.976387417681569337617621359592, 9.609426221544422366479833805124, 10.72139201898917303817002564827

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