↑

Properties

Label 2-240-15.2-c1-0-1
Degree $2$
Conductor $240$
Sign $-0.391 - 0.920i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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 + 1.41i)3-s + (1 + 2i)5-s + (−0.414 − 0.414i)7-s + (−1.00 − 2.82i)9-s + 4.82i·11-s + (−1.82 + 1.82i)13-s + (−3.82 − 0.585i)15-s + (−3.82 + 3.82i)17-s − 4.82i·19-s + (1 − 0.171i)21-s + (1.58 + 1.58i)23-s + (−3 + 4i)25-s + (5.00 + 1.41i)27-s + 7.65·29-s + 5.65·31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + (0.447 + 0.894i)5-s + (−0.156 − 0.156i)7-s + (−0.333 − 0.942i)9-s + 1.45i·11-s + (−0.507 + 0.507i)13-s + (−0.988 − 0.151i)15-s + (−0.928 + 0.928i)17-s − 1.10i·19-s + (0.218 − 0.0374i)21-s + (0.330 + 0.330i)23-s + (−0.600 + 0.800i)25-s + (0.962 + 0.272i)27-s + 1.42·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.391 - 0.920i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528680 + 0.799024i\)
\(L(\frac12)\) \(\approx\) \(0.528680 + 0.799024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1 - 1.41i)T \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 + (0.414 + 0.414i)T + 7iT^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + (1.82 - 1.82i)T - 13iT^{2} \)
17 \( 1 + (3.82 - 3.82i)T - 17iT^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (-1.58 - 1.58i)T + 23iT^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (0.171 + 0.171i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2.41 - 2.41i)T - 43iT^{2} \)
47 \( 1 + (-6.41 + 6.41i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (-4.07 - 4.07i)T + 67iT^{2} \)
71 \( 1 - 6.48iT - 71T^{2} \)
73 \( 1 + (-6.65 + 6.65i)T - 73iT^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 + (5.24 + 5.24i)T + 83iT^{2} \)
89 \( 1 + 4.34T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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

−12.22874148580091925975493762269, −11.34127138126336127988795485988, −10.36185647206653693729868290610, −9.867818722441321499550516902177, −8.829071249201966031866485214267, −7.04321969015559544984800931797, −6.52130409254316779265758767795, −5.06975492303816037231165127328, −4.07976335334290891779966422438, −2.44633282563865214618409515269, 0.849697606789767347319049905355, 2.67274229363317024227521916753, 4.75237944573905987417453971246, 5.74440843936315810179120356496, 6.56522091801796313853129072653, 8.007464440794589597600333635888, 8.688698113463902045504923939393, 9.955005065899607055799268894553, 11.04696859900534597233339511088, 12.00003287636609261388784649831

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