↑

Properties

Label 2-603-67.29-c1-0-15
Degree $2$
Conductor $603$
Sign $0.667 + 0.744i$
Analytic cond. $4.81497$
Root an. cond. $2.19430$
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  + (−0.809 + 1.40i)2-s + (−0.309 − 0.535i)4-s − 1.23·5-s + (0.118 + 0.204i)7-s − 2.23·8-s + (1.00 − 1.73i)10-s + (−2.5 − 4.33i)11-s + (−0.118 + 0.204i)13-s − 0.381·14-s + (2.42 − 4.20i)16-s + (1.11 − 1.93i)17-s + (0.881 − 1.52i)19-s + (0.381 + 0.661i)20-s + 8.09·22-s + (−0.881 + 1.52i)23-s + ⋯
L(s)  = 1  + (−0.572 + 0.990i)2-s + (−0.154 − 0.267i)4-s − 0.552·5-s + (0.0446 + 0.0772i)7-s − 0.790·8-s + (0.316 − 0.547i)10-s + (−0.753 − 1.30i)11-s + (−0.0327 + 0.0567i)13-s − 0.102·14-s + (0.606 − 1.05i)16-s + (0.271 − 0.469i)17-s + (0.202 − 0.350i)19-s + (0.0854 + 0.147i)20-s + 1.72·22-s + (−0.183 + 0.318i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ 0.667 + 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417647 - 0.186578i\)
\(L(\frac12)\) \(\approx\) \(0.417647 - 0.186578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 + (8 + 1.73i)T \)
good2 \( 1 + (0.809 - 1.40i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 + (-0.118 - 0.204i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.118 - 0.204i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.11 + 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.881 + 1.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.881 - 1.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.263 - 0.457i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.35 + 5.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.73 + 6.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.11 + 3.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.763T + 43T^{2} \)
47 \( 1 + (-0.881 - 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.70T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + (6.35 - 11.0i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-2.26 - 3.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.35 - 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.73 + 8.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.5 + 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.45660843292682918832121723163, −9.309838160015493808318879971949, −8.635179699240397898978634675524, −7.74585503417883854332187201537, −7.31518926984035295054711976081, −6.02788215423722844460894303279, −5.42663264471492338075285989315, −3.84834682310878290895221448941, −2.73782668027411919950180182284, −0.30335019161390974118138731185, 1.55040181354472491143208355096, 2.72668659767645829097105305415, 3.90964210761127025635046268190, 5.09377937037315435175328445002, 6.31460805488209673941624770385, 7.49910278598997704706682832338, 8.227993447605974119188644389324, 9.303645157885287244499378371114, 10.11093180744591632424212826559, 10.59527013311302741405806923433

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