Properties

Label 2-1875-75.11-c0-0-0
Degree $2$
Conductor $1875$
Sign $-0.684 - 0.728i$
Analytic cond. $0.935746$
Root an. cond. $0.967340$
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  + (0.951 + 1.30i)2-s + (−0.951 − 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (0.951 − 1.30i)12-s + (−0.587 + 0.190i)17-s + 1.61i·18-s + (0.5 + 1.53i)19-s + (0.363 + 0.5i)23-s + 24-s + (−0.587 − 0.809i)27-s + (0.190 + 0.587i)31-s + 0.999i·32-s + ⋯
L(s)  = 1  + (0.951 + 1.30i)2-s + (−0.951 − 0.309i)3-s + (−0.500 + 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (0.951 − 1.30i)12-s + (−0.587 + 0.190i)17-s + 1.61i·18-s + (0.5 + 1.53i)19-s + (0.363 + 0.5i)23-s + 24-s + (−0.587 − 0.809i)27-s + (0.190 + 0.587i)31-s + 0.999i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.684 - 0.728i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ -0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347533203\)
\(L(\frac12)\) \(\approx\) \(1.347533203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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

−9.758925077369481245649985174083, −8.479281200206534524959669445047, −7.82626339446573046880172265159, −7.04606161582715253151608513188, −6.46877208738685043164949372309, −5.70776239995003194994851274722, −5.14778057676956163731895621354, −4.30592670124311872824530189842, −3.40884176163966516618573501749, −1.63510527933066331543180380800, 0.892096803552340349464468896674, 2.26925028582572100117480712534, 3.24839790388749761311368701950, 4.26809131163049349859133479934, 4.83705870552929864618471692874, 5.51473185852872656977541752200, 6.51065258157281094273876591409, 7.26579751573340735848435728097, 8.639538831052658461198103849226, 9.619762096335287619548405915966

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