↑

Properties

Label 2-722-361.39-c1-0-21
Degree $2$
Conductor $722$
Sign $0.211 - 0.977i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
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  + (0.401 + 0.915i)2-s + (2.24 + 1.74i)3-s + (−0.677 + 0.735i)4-s + (−0.330 − 3.98i)5-s + (−0.698 + 2.75i)6-s + (2.23 + 2.42i)7-s + (−0.945 − 0.324i)8-s + (1.24 + 4.93i)9-s + (3.51 − 1.90i)10-s + (0.266 − 1.05i)11-s + (−2.80 + 0.468i)12-s + (0.961 + 0.748i)13-s + (−1.32 + 3.01i)14-s + (6.22 − 9.52i)15-s + (−0.0825 − 0.996i)16-s + (2.57 + 2.79i)17-s + ⋯
L(s)  = 1  + (0.284 + 0.647i)2-s + (1.29 + 1.00i)3-s + (−0.338 + 0.367i)4-s + (−0.147 − 1.78i)5-s + (−0.285 + 1.12i)6-s + (0.843 + 0.915i)7-s + (−0.334 − 0.114i)8-s + (0.416 + 1.64i)9-s + (1.11 − 0.601i)10-s + (0.0802 − 0.316i)11-s + (−0.809 + 0.135i)12-s + (0.266 + 0.207i)13-s + (−0.353 + 0.805i)14-s + (1.60 − 2.45i)15-s + (−0.0206 − 0.249i)16-s + (0.624 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13919 + 1.72645i\)
\(L(\frac12)\) \(\approx\) \(2.13919 + 1.72645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.401 - 0.915i)T \)
19 \( 1 + (-2.49 + 3.57i)T \)
good3 \( 1 + (-2.24 - 1.74i)T + (0.736 + 2.90i)T^{2} \)
5 \( 1 + (0.330 + 3.98i)T + (-4.93 + 0.822i)T^{2} \)
7 \( 1 + (-2.23 - 2.42i)T + (-0.578 + 6.97i)T^{2} \)
11 \( 1 + (-0.266 + 1.05i)T + (-9.67 - 5.23i)T^{2} \)
13 \( 1 + (-0.961 - 0.748i)T + (3.19 + 12.6i)T^{2} \)
17 \( 1 + (-2.57 - 2.79i)T + (-1.40 + 16.9i)T^{2} \)
23 \( 1 + (2.73 - 2.13i)T + (5.64 - 22.2i)T^{2} \)
29 \( 1 + (-0.0715 - 0.0777i)T + (-2.39 + 28.9i)T^{2} \)
31 \( 1 + (3.16 - 7.22i)T + (-20.9 - 22.8i)T^{2} \)
37 \( 1 + (0.651 - 2.57i)T + (-32.5 - 17.6i)T^{2} \)
41 \( 1 + (-5.85 + 8.96i)T + (-16.4 - 37.5i)T^{2} \)
43 \( 1 + (8.63 + 4.67i)T + (23.5 + 35.9i)T^{2} \)
47 \( 1 + (-1.08 + 4.27i)T + (-41.3 - 22.3i)T^{2} \)
53 \( 1 + (1.48 - 5.85i)T + (-46.6 - 25.2i)T^{2} \)
59 \( 1 + (-1.85 + 2.84i)T + (-23.7 - 54.0i)T^{2} \)
61 \( 1 + (-3.98 + 1.36i)T + (48.1 - 37.4i)T^{2} \)
67 \( 1 + (8.62 + 2.96i)T + (52.8 + 41.1i)T^{2} \)
71 \( 1 + (13.6 + 4.69i)T + (56.0 + 43.6i)T^{2} \)
73 \( 1 + (-4.88 - 5.30i)T + (-6.02 + 72.7i)T^{2} \)
79 \( 1 + (12.0 + 6.50i)T + (43.2 + 66.1i)T^{2} \)
83 \( 1 + (-0.903 + 10.9i)T + (-81.8 - 13.6i)T^{2} \)
89 \( 1 + (3.27 - 3.56i)T + (-7.34 - 88.6i)T^{2} \)
97 \( 1 + (6.92 - 2.37i)T + (76.5 - 59.5i)T^{2} \)
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

−10.24751712918425554821209490854, −9.180243247828126652733303543803, −8.776324587361732971048139092548, −8.386018343914403478048533443604, −7.53849493941284049375844762905, −5.66720600827792632422174149471, −5.07674743977854620256920612718, −4.29788907750393826962136161226, −3.36017689873700191972536284987, −1.74976293130596857450380565559, 1.44477243574562942317458879580, 2.54134492852320822898955296135, 3.31386857508667928669571686459, 4.18519233833350506446283082497, 5.97313742323384374198167029638, 7.03931786690293825490486154333, 7.63414549080139818235608448889, 8.158526795994669622558705150340, 9.621636767104855160084580088886, 10.20592421909569927209885589089

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