Properties

Label 2-1200-15.8-c1-0-3
Degree $2$
Conductor $1200$
Sign $-0.973 - 0.229i$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
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  + (1.22 + 1.22i)3-s + (−2.44 + 2.44i)7-s + 2.99i·9-s + (−4.89 − 4.89i)13-s + 8i·19-s − 5.99·21-s + (−3.67 + 3.67i)27-s − 4·31-s + (−4.89 + 4.89i)37-s − 11.9i·39-s + (−7.34 − 7.34i)43-s − 4.99i·49-s + (−9.79 + 9.79i)57-s + 14·61-s + (−7.34 − 7.34i)63-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (−0.925 + 0.925i)7-s + 0.999i·9-s + (−1.35 − 1.35i)13-s + 1.83i·19-s − 1.30·21-s + (−0.707 + 0.707i)27-s − 0.718·31-s + (−0.805 + 0.805i)37-s − 1.92i·39-s + (−1.12 − 1.12i)43-s − 0.714i·49-s + (−1.29 + 1.29i)57-s + 1.79·61-s + (−0.925 − 0.925i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9677605900\)
\(L(\frac12)\) \(\approx\) \(0.9677605900\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (2.44 - 2.44i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (4.89 + 4.89i)T + 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.34 + 7.34i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (2.44 - 2.44i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-9.79 - 9.79i)T + 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (9.79 - 9.79i)T - 97iT^{2} \)
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

−10.02922069743302455057424238589, −9.469577786801670800668305069568, −8.488734964847684474648142344751, −7.928646102781088426641204299211, −6.90226927090688412518711403268, −5.62640547881428672876678861360, −5.19937805363465350268584380509, −3.78715910760486859992191564319, −3.06408519491111952680578615938, −2.14241736159813608414000170484, 0.34629963828253387581457182745, 1.97157040146593949985367041054, 2.97525660020543560451748516251, 3.97674849085225271863939745011, 4.95108372630036333510499726638, 6.46906986043415664692664949963, 7.00605239952626062650900349976, 7.42735976743338936946751589829, 8.651565514601144550104241192966, 9.417016290381244423872452984419

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