↑

Properties

Label 2-240-5.3-c2-0-10
Degree $2$
Conductor $240$
Sign $-0.484 + 0.874i$
Analytic cond. $6.53952$
Root an. cond. $2.55724$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (−3.36 + 3.70i)5-s + (8.78 − 8.78i)7-s + 2.99i·9-s − 13.7·11-s + (−4.88 − 4.88i)13-s + (8.65 − 0.413i)15-s + (5.99 − 5.99i)17-s − 25.5i·19-s − 21.5·21-s + (−18.4 − 18.4i)23-s + (−2.38 − 24.8i)25-s + (3.67 − 3.67i)27-s − 37.2i·29-s − 31.6·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.672 + 0.740i)5-s + (1.25 − 1.25i)7-s + 0.333i·9-s − 1.24·11-s + (−0.376 − 0.376i)13-s + (0.576 − 0.0275i)15-s + (0.352 − 0.352i)17-s − 1.34i·19-s − 1.02·21-s + (−0.801 − 0.801i)23-s + (−0.0954 − 0.995i)25-s + (0.136 − 0.136i)27-s − 1.28i·29-s − 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(6.53952\)
Root analytic conductor: \(2.55724\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1),\ -0.484 + 0.874i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.438753 - 0.744555i\)
\(L(\frac12)\) \(\approx\) \(0.438753 - 0.744555i\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (3.36 - 3.70i)T \)
good7 \( 1 + (-8.78 + 8.78i)T - 49iT^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + (4.88 + 4.88i)T + 169iT^{2} \)
17 \( 1 + (-5.99 + 5.99i)T - 289iT^{2} \)
19 \( 1 + 25.5iT - 361T^{2} \)
23 \( 1 + (18.4 + 18.4i)T + 529iT^{2} \)
29 \( 1 + 37.2iT - 841T^{2} \)
31 \( 1 + 31.6T + 961T^{2} \)
37 \( 1 + (32.5 - 32.5i)T - 1.36e3iT^{2} \)
41 \( 1 - 36.7T + 1.68e3T^{2} \)
43 \( 1 + (-24.5 - 24.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (20.4 - 20.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-33.5 - 33.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 7.54iT - 3.48e3T^{2} \)
61 \( 1 - 43.6T + 3.72e3T^{2} \)
67 \( 1 + (-60.1 + 60.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 18.1T + 5.04e3T^{2} \)
73 \( 1 + (-68.8 - 68.8i)T + 5.32e3iT^{2} \)
79 \( 1 - 22.2iT - 6.24e3T^{2} \)
83 \( 1 + (-0.00221 - 0.00221i)T + 6.88e3iT^{2} \)
89 \( 1 - 77.1iT - 7.92e3T^{2} \)
97 \( 1 + (-29.4 + 29.4i)T - 9.40e3iT^{2} \)
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

−11.30847238380998906772460822528, −10.88588832534291440670694978034, −10.04351968483199562741856927923, −8.099137840186890544166418387604, −7.66630954159683000512470500709, −6.80461879490119162293253304890, −5.22068201941008033847045217495, −4.22954568454200379604459161996, −2.53002479900066463083242202422, −0.47616650462900462278493896198, 1.89213658998693001076115331171, 3.84535539443648111771279090742, 5.25538483456982056085558099934, 5.49607639958100537950037950085, 7.56870774786185816091032715495, 8.295876160091933864089700770763, 9.181924547788601406513766904233, 10.45297203688039237205614614001, 11.39066739724860132857919559369, 12.18068947884060319655261976695

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