↑

Properties

Label 2-2400-24.11-c1-0-12
Degree $2$
Conductor $2400$
Sign $0.816 - 0.577i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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 + (−1.00 + 2.82i)9-s − 2.82i·11-s + 5.65i·17-s − 2·19-s + (5.00 − 1.41i)27-s + (−4.00 + 2.82i)33-s + 11.3i·41-s − 10·43-s + 7·49-s + (8.00 − 5.65i)51-s + (2 + 2.82i)57-s + 14.1i·59-s + 14·67-s − 2·73-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + (−0.333 + 0.942i)9-s − 0.852i·11-s + 1.37i·17-s − 0.458·19-s + (0.962 − 0.272i)27-s + (−0.696 + 0.492i)33-s + 1.76i·41-s − 1.52·43-s + 49-s + (1.12 − 0.792i)51-s + (0.264 + 0.374i)57-s + 1.84i·59-s + 1.71·67-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.017906230\)
\(L(\frac12)\) \(\approx\) \(1.017906230\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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

−8.702997871714932983765547736546, −8.298527165408829152969619472751, −7.50202561122533516200348335277, −6.54549270087613272619010481884, −6.08244806334364017350746382929, −5.29915590012214749850616505894, −4.28684841584362602698475383144, −3.20878852499132098605088769285, −2.07390533808297228106569892821, −1.02709656310998515454024541371, 0.43629136808868055354532618154, 2.09261026176567940835496866639, 3.26141843624389429733080103779, 4.17880777761139982658805710114, 4.95976962626720630007075593800, 5.53048650246419820712533709257, 6.64004200905974742111765148397, 7.13742115109991521178988712770, 8.228370084764894638761475578086, 9.107897721582570851156740561348

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