Properties

Label 2-2576-2576.1637-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.405 + 0.914i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (0.281 − 0.959i)2-s + (−0.841 − 0.540i)4-s + (0.989 − 0.142i)7-s + (−0.755 + 0.654i)8-s + (−0.540 − 0.841i)9-s + (1.64 − 0.613i)11-s + (0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)18-s + (−0.125 − 1.75i)22-s + (0.841 + 0.540i)23-s + (−0.909 + 0.415i)25-s + (−0.909 − 0.415i)28-s + (−0.697 − 0.0498i)29-s + (0.989 − 0.142i)32-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.841 − 0.540i)4-s + (0.989 − 0.142i)7-s + (−0.755 + 0.654i)8-s + (−0.540 − 0.841i)9-s + (1.64 − 0.613i)11-s + (0.142 − 0.989i)14-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)18-s + (−0.125 − 1.75i)22-s + (0.841 + 0.540i)23-s + (−0.909 + 0.415i)25-s + (−0.909 − 0.415i)28-s + (−0.697 − 0.0498i)29-s + (0.989 − 0.142i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.405 + 0.914i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.405 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444338466\)
\(L(\frac12)\) \(\approx\) \(1.444338466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.281 + 0.959i)T \)
7 \( 1 + (-0.989 + 0.142i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
good3 \( 1 + (0.540 + 0.841i)T^{2} \)
5 \( 1 + (0.909 - 0.415i)T^{2} \)
11 \( 1 + (-1.64 + 0.613i)T + (0.755 - 0.654i)T^{2} \)
13 \( 1 + (0.281 - 0.959i)T^{2} \)
17 \( 1 + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.989 - 0.142i)T^{2} \)
29 \( 1 + (0.697 + 0.0498i)T + (0.989 + 0.142i)T^{2} \)
31 \( 1 + (-0.841 - 0.540i)T^{2} \)
37 \( 1 + (-1.56 - 0.340i)T + (0.909 + 0.415i)T^{2} \)
41 \( 1 + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (1.71 + 0.936i)T + (0.540 + 0.841i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.12 + 1.50i)T + (-0.281 - 0.959i)T^{2} \)
59 \( 1 + (-0.281 + 0.959i)T^{2} \)
61 \( 1 + (-0.540 + 0.841i)T^{2} \)
67 \( 1 + (1.86 + 0.697i)T + (0.755 + 0.654i)T^{2} \)
71 \( 1 + (1.74 - 0.797i)T + (0.654 - 0.755i)T^{2} \)
73 \( 1 + (-0.142 - 0.989i)T^{2} \)
79 \( 1 + (-0.258 + 1.80i)T + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.909 + 0.415i)T^{2} \)
89 \( 1 + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 - 0.909i)T^{2} \)
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

−8.957898735793961746149824980857, −8.433131245695395835675646818628, −7.33705957167642737564609980349, −6.27014586215400434586018161339, −5.65695161082939997784142569871, −4.69732258411797692542938645103, −3.80677058682750365488867062997, −3.28725010791056216184967421664, −1.90449532608987751103858892681, −1.01071570683441398928850271546, 1.53206950379501914712460949562, 2.81534166784020889170301130949, 4.17935315464964523368353216399, 4.53823018697961846710763811260, 5.51515147088176961390717368561, 6.17819182472517495032097091543, 7.09710508262888529231464723882, 7.71511624823272808870711573084, 8.455622751254460199758389691221, 9.045570671927046839346323330488

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