↑

Properties

Label 2-240-15.8-c1-0-8
Degree $2$
Conductor $240$
Sign $0.607 + 0.794i$
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  + (0.618 − 1.61i)3-s + 2.23·5-s + (1 − i)7-s + (−2.23 − 2.00i)9-s + 4.47i·11-s + (−3 − 3i)13-s + (1.38 − 3.61i)15-s + (2.23 + 2.23i)17-s − 2i·19-s + (−1 − 2.23i)21-s + (2.23 − 2.23i)23-s + 5.00·25-s + (−4.61 + 2.38i)27-s − 4.47·29-s − 4·31-s + ⋯
L(s)  = 1  + (0.356 − 0.934i)3-s + 0.999·5-s + (0.377 − 0.377i)7-s + (−0.745 − 0.666i)9-s + 1.34i·11-s + (−0.832 − 0.832i)13-s + (0.356 − 0.934i)15-s + (0.542 + 0.542i)17-s − 0.458i·19-s + (−0.218 − 0.487i)21-s + (0.466 − 0.466i)23-s + 1.00·25-s + (−0.888 + 0.458i)27-s − 0.830·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.607 + 0.794i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40138 - 0.692953i\)
\(L(\frac12)\) \(\approx\) \(1.40138 - 0.692953i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.61i)T \)
5 \( 1 - 2.23T \)
good7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-2.23 - 2.23i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-2.23 + 2.23i)T - 23iT^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (-6.70 - 6.70i)T + 47iT^{2} \)
53 \( 1 + (-2.23 + 2.23i)T - 53iT^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-1 + i)T - 67iT^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (6.70 - 6.70i)T - 83iT^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (-9 + 9i)T - 97iT^{2} \)
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

−12.46180928359820454140786385982, −10.99467271289138221075067801193, −9.963716604656607285115406418372, −9.133371227550648904924832666472, −7.81342454328930457337331884930, −7.13510109905449442202161937971, −5.96885691089216655569722561515, −4.77540700702460406257143055254, −2.81366727372100398059723332847, −1.57657519551515762396399213495, 2.23804720887372384460904940647, 3.60271592086113855295189830632, 5.16813958803800113904499677103, 5.77208246777687827150158630480, 7.38320574069158203054610038036, 8.806836119434690603895943353598, 9.237980633035091110644984590471, 10.29585846296444545320743752773, 11.12988092235765997920627385626, 12.13115066161320219377902936827

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