↑

Properties

Label 2-1800-120.107-c0-0-2
Degree $2$
Conductor $1800$
Sign $-0.662 + 0.749i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1 + i)7-s + (0.707 − 0.707i)8-s − 1.41i·11-s + (−1 − i)13-s + 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + 1.41i·26-s + (−1.00 − 1.00i)28-s + (0.707 + 0.707i)32-s + (1 − i)37-s − 1.41i·41-s + 1.41·44-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−1 + i)7-s + (0.707 − 0.707i)8-s − 1.41i·11-s + (−1 − i)13-s + 1.41·14-s − 1.00·16-s + (−1.00 + 1.00i)22-s + 1.41i·26-s + (−1.00 − 1.00i)28-s + (0.707 + 0.707i)32-s + (1 − i)37-s − 1.41i·41-s + 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4514187742\)
\(L(\frac12)\) \(\approx\) \(0.4514187742\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
53 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + iT^{2} \)
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

−9.191014667405723660504795430837, −8.598973260129755903828781729900, −7.86889063283811393664356941170, −6.92749616152263268203384094477, −5.96493263345832002090560928671, −5.20712404825805782848894420757, −3.72497869127543524949128690772, −3.02069625978704744776332808264, −2.26854763488013718592508395628, −0.42656908383430694952512642732, 1.47177263539292877161693066196, 2.75230505357438377983383660763, 4.33756928787798329008321655546, 4.72972390727750369133384507388, 6.11802112009196194063730371145, 6.79025914702659864312930594606, 7.30467648413348877852331888392, 7.964716171337742852493463201894, 9.190799493176753233864810852749, 9.763385772627527490390965589519

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