↑

Properties

Label 2-240-60.59-c1-0-5
Degree $2$
Conductor $240$
Sign $0.700 - 0.713i$
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  + (1.41 + i)3-s + (1.73 + 1.41i)5-s + (1.00 + 2.82i)9-s − 4.89·11-s − 4.89i·13-s + (1.03 + 3.73i)15-s + 3.46·17-s + 3.46i·19-s − 6i·23-s + (0.999 + 4.89i)25-s + (−1.41 + 5.00i)27-s − 2.82i·29-s − 3.46i·31-s + (−6.92 − 4.89i)33-s + 4.89i·37-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + (0.774 + 0.632i)5-s + (0.333 + 0.942i)9-s − 1.47·11-s − 1.35i·13-s + (0.267 + 0.963i)15-s + 0.840·17-s + 0.794i·19-s − 1.25i·23-s + (0.199 + 0.979i)25-s + (−0.272 + 0.962i)27-s − 0.525i·29-s − 0.622i·31-s + (−1.20 − 0.852i)33-s + 0.805i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.700 - 0.713i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.700 - 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54747 + 0.648974i\)
\(L(\frac12)\) \(\approx\) \(1.54747 + 0.648974i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 4.89iT - 37T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 - 9.79T + 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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

−12.53113279786017960484543492675, −10.73906093052727336084202626502, −10.36072663696393780395411476805, −9.590944626510036200877878506744, −8.195996805270922995381470956289, −7.62904820949197580670660517355, −5.97236134116032492543438736585, −5.01540522841742664948227822709, −3.31710489518452405877237631787, −2.40831555262919132870408939420, 1.64064815522268122915659503232, 2.95941367850391128428752074999, 4.68620567307197295270987610537, 5.89872408412967296332443677894, 7.16829209880076257687818841995, 8.086970329360573443225722929276, 9.167335236279787324987815983951, 9.729634770237274410311172636080, 11.06793501026578846280900705146, 12.36099509825797163235335399198

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