↑

Properties

Label 2-2576-2576.1707-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.366 + 0.930i$
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  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (0.909 − 0.415i)7-s + (−0.540 − 0.841i)8-s + (0.989 − 0.142i)9-s + (−1.75 − 0.956i)11-s + (0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−1.94 + 0.424i)22-s + (0.142 − 0.989i)23-s + (−0.281 + 0.959i)25-s + (−0.281 − 0.959i)28-s + (−0.203 + 0.936i)29-s + (−0.909 + 0.415i)32-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (0.909 − 0.415i)7-s + (−0.540 − 0.841i)8-s + (0.989 − 0.142i)9-s + (−1.75 − 0.956i)11-s + (0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−1.94 + 0.424i)22-s + (0.142 − 0.989i)23-s + (−0.281 + 0.959i)25-s + (−0.281 − 0.959i)28-s + (−0.203 + 0.936i)29-s + (−0.909 + 0.415i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.366 + 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.887130046\)
\(L(\frac12)\) \(\approx\) \(1.887130046\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (-0.909 + 0.415i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (-0.989 + 0.142i)T^{2} \)
5 \( 1 + (0.281 - 0.959i)T^{2} \)
11 \( 1 + (1.75 + 0.956i)T + (0.540 + 0.841i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.909 + 0.415i)T^{2} \)
29 \( 1 + (0.203 - 0.936i)T + (-0.909 - 0.415i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (-0.559 - 0.418i)T + (0.281 + 0.959i)T^{2} \)
41 \( 1 + (-0.959 - 0.281i)T^{2} \)
43 \( 1 + (-1.19 + 0.0855i)T + (0.989 - 0.142i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.898 + 0.334i)T + (0.755 + 0.654i)T^{2} \)
59 \( 1 + (0.755 - 0.654i)T^{2} \)
61 \( 1 + (0.989 + 0.142i)T^{2} \)
67 \( 1 + (0.373 - 0.203i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (-1.45 - 0.425i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.281 - 0.959i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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

−8.916612916934173551554640006553, −7.967658103827404027132850493295, −7.34126933981103150583541525692, −6.38202757260002961229278901965, −5.38437623749707233012944911745, −4.90914982945940416637843005010, −4.04889484306631948038330719061, −3.11774318223356343521370731075, −2.14643682771766246653311842430, −1.00523807211973354301400007093, 1.97045617944314290018837540296, 2.66290253591412511337047323557, 4.05265493258331533919899650499, 4.69660398133479845186535893545, 5.30009152806888578807339688974, 6.08003702758377403297666887906, 7.17726398242352114779816405202, 7.79452309382326433378347989075, 8.029644619330224910667088346864, 9.226193145628431093953011091851

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