↑

Properties

Label 2-4620-385.384-c1-0-60
Degree $2$
Conductor $4620$
Sign $0.803 + 0.595i$
Analytic cond. $36.8908$
Root an. cond. $6.07378$
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  − 3-s + (1.56 + 1.59i)5-s + (−0.492 + 2.59i)7-s + 9-s + (−1.11 − 3.12i)11-s + 4.71i·13-s + (−1.56 − 1.59i)15-s − 4.60i·17-s + 3.62·19-s + (0.492 − 2.59i)21-s − 8.38i·23-s + (−0.0720 + 4.99i)25-s − 27-s − 5.64i·29-s − 5.75i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.701 + 0.712i)5-s + (−0.186 + 0.982i)7-s + 0.333·9-s + (−0.334 − 0.942i)11-s + 1.30i·13-s + (−0.405 − 0.411i)15-s − 1.11i·17-s + 0.831·19-s + (0.107 − 0.567i)21-s − 1.74i·23-s + (−0.0144 + 0.999i)25-s − 0.192·27-s − 1.04i·29-s − 1.03i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4620\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.803 + 0.595i$
Analytic conductor: \(36.8908\)
Root analytic conductor: \(6.07378\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4620} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4620,\ (\ :1/2),\ 0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417731172\)
\(L(\frac12)\) \(\approx\) \(1.417731172\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (-1.56 - 1.59i)T \)
7 \( 1 + (0.492 - 2.59i)T \)
11 \( 1 + (1.11 + 3.12i)T \)
good13 \( 1 - 4.71iT - 13T^{2} \)
17 \( 1 + 4.60iT - 17T^{2} \)
19 \( 1 - 3.62T + 19T^{2} \)
23 \( 1 + 8.38iT - 23T^{2} \)
29 \( 1 + 5.64iT - 29T^{2} \)
31 \( 1 + 5.75iT - 31T^{2} \)
37 \( 1 + 1.86iT - 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 2.68T + 43T^{2} \)
47 \( 1 - 5.10T + 47T^{2} \)
53 \( 1 + 7.50iT - 53T^{2} \)
59 \( 1 + 12.9iT - 59T^{2} \)
61 \( 1 - 9.36T + 61T^{2} \)
67 \( 1 - 0.875iT - 67T^{2} \)
71 \( 1 + 1.64T + 71T^{2} \)
73 \( 1 - 9.03iT - 73T^{2} \)
79 \( 1 - 1.69iT - 79T^{2} \)
83 \( 1 + 6.50iT - 83T^{2} \)
89 \( 1 + 8.66iT - 89T^{2} \)
97 \( 1 - 0.227T + 97T^{2} \)
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

−8.355481699916011252855920622381, −7.30578948094337052313723930582, −6.60183111192105787796716889693, −6.12111778300842903252391332354, −5.42463673795572648940356716621, −4.73735429322476391336326563601, −3.58255875707852726886588941980, −2.61654405625105220467315033260, −2.05332772717752294613645334913, −0.47473987211469075388951119332, 1.04012079907792439429651806307, 1.66202528111318030042944642271, 3.09625977933739093265546822452, 3.92250545980949277804231918834, 4.93131565021696354618117681740, 5.36247440942293547314060850036, 6.05912266418330463746762067291, 7.04958152110288921437519103981, 7.52565394914623226205469280536, 8.325291610296132079045253026270

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