↑

Properties

Label 2-2523-87.62-c0-0-2
Degree $2$
Conductor $2523$
Sign $0.545 - 0.838i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (1.45 − 0.702i)7-s + (−0.623 + 0.781i)9-s + i·12-s + (−0.385 − 0.483i)13-s + (0.623 + 0.781i)16-s + (−0.268 + 0.556i)19-s + (1.26 + 1.00i)21-s + (−0.900 − 0.433i)25-s + (−0.974 − 0.222i)27-s + 1.61·28-s + (−1.57 − 0.360i)31-s + (−0.900 + 0.433i)36-s + (0.483 + 0.385i)37-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (0.900 + 0.433i)4-s + (1.45 − 0.702i)7-s + (−0.623 + 0.781i)9-s + i·12-s + (−0.385 − 0.483i)13-s + (0.623 + 0.781i)16-s + (−0.268 + 0.556i)19-s + (1.26 + 1.00i)21-s + (−0.900 − 0.433i)25-s + (−0.974 − 0.222i)27-s + 1.61·28-s + (−1.57 − 0.360i)31-s + (−0.900 + 0.433i)36-s + (0.483 + 0.385i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910545797\)
\(L(\frac12)\) \(\approx\) \(1.910545797\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.433 - 0.900i)T \)
29 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (-1.45 + 0.702i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.385 + 0.483i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.268 - 0.556i)T + (-0.623 - 0.781i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (1.57 + 0.360i)T + (0.900 + 0.433i)T^{2} \)
37 \( 1 + (-0.483 - 0.385i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.57 + 0.360i)T + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.385 - 0.483i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + (0.483 + 0.385i)T + (0.222 + 0.974i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.900 - 0.433i)T^{2} \)
97 \( 1 + (0.702 - 1.45i)T + (-0.623 - 0.781i)T^{2} \)
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

−9.198741683923381934064517485756, −8.175225921791236527592080602077, −7.86217716353728188834075505382, −7.24513916819727191492124541488, −5.99397253433241831699096892844, −5.22476412893152146502880903194, −4.26032678060418696595952493445, −3.69453842871731297251181803154, −2.54513133896209034462439223399, −1.71639290035704922173673165775, 1.41706068632629056491045422042, 2.08278313653033799726172452609, 2.78914584896199585818192949806, 4.17667941592646577487040086916, 5.36023088290324108303957988621, 5.84429098802230407598715495907, 6.83365807265813801706048896330, 7.50219833522395399168067400899, 7.985173233930418166042860381410, 8.958609519761347029175382126358

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