↑

Properties

Label 2-605-55.49-c1-0-27
Degree $2$
Conductor $605$
Sign $0.803 - 0.595i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (2.48 − 0.805i)2-s + (−1.25 + 1.73i)3-s + (3.88 − 2.82i)4-s + (0.0494 + 2.23i)5-s + (−1.72 + 5.30i)6-s + (0.581 + 0.800i)7-s + (4.29 − 5.90i)8-s + (−0.487 − 1.50i)9-s + (1.92 + 5.50i)10-s + 10.2i·12-s + (−2.92 + 0.950i)13-s + (2.08 + 1.51i)14-s + (−3.93 − 2.72i)15-s + (2.91 − 8.98i)16-s + (1.53 + 0.500i)17-s + (−2.41 − 3.33i)18-s + ⋯
L(s)  = 1  + (1.75 − 0.569i)2-s + (−0.726 + 0.999i)3-s + (1.94 − 1.41i)4-s + (0.0220 + 0.999i)5-s + (−0.703 + 2.16i)6-s + (0.219 + 0.302i)7-s + (1.51 − 2.08i)8-s + (−0.162 − 0.500i)9-s + (0.608 + 1.74i)10-s + 2.96i·12-s + (−0.811 + 0.263i)13-s + (0.557 + 0.405i)14-s + (−1.01 − 0.703i)15-s + (0.729 − 2.24i)16-s + (0.373 + 0.121i)17-s + (−0.570 − 0.785i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.803 - 0.595i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11615 + 1.02809i\)
\(L(\frac12)\) \(\approx\) \(3.11615 + 1.02809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0494 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (-2.48 + 0.805i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.25 - 1.73i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-0.581 - 0.800i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.92 - 0.950i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.53 - 0.500i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.32 - 3.86i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 + (-5.91 + 4.29i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.816 + 2.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.64 + 2.26i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.12 + 0.814i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + (1.25 - 1.73i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.26 - 0.734i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.90 + 7.19i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.309 - 0.953i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + (0.0753 - 0.231i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.03 + 11.0i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.37 + 7.31i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.66 + 1.84i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (12.1 - 3.93i)T + (78.4 - 57.0i)T^{2} \)
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

−10.89544289780971214437552753220, −10.26847751734192490191211351723, −9.712567481116300613137953185693, −7.75235249356591268280540818400, −6.61002183347315509473044487515, −5.80625044710269090251532276543, −5.05771656719307507792065797129, −4.21974142733857134802546957618, −3.27189219735755668287175214461, −2.19938166204721064877265879609, 1.31181441047361174517263942644, 2.97076902538706567260300227170, 4.32212036275713663803726468778, 5.32151286672138104240495630455, 5.60073982857600704303926114071, 7.02405626718892057226817317656, 7.23423407472784378689995087398, 8.335794371661934507385997893592, 9.772101099701468089632353914835, 11.26081876244225198082036254799

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