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Properties

Label 2-603-67.29-c1-0-21
Degree $2$
Conductor $603$
Sign $-0.456 + 0.889i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 − 2.14i)2-s + (−2.05 − 3.56i)4-s + 2.75·5-s + (2.34 + 4.06i)7-s − 5.22·8-s + (3.40 − 5.89i)10-s + (−2.47 − 4.28i)11-s + (0.0567 − 0.0983i)13-s + 11.6·14-s + (−2.34 + 4.06i)16-s + (2.61 − 4.52i)17-s + (−0.209 + 0.363i)19-s + (−5.66 − 9.80i)20-s − 12.2·22-s + (−1.81 + 3.14i)23-s + ⋯
L(s)  = 1  + (0.874 − 1.51i)2-s + (−1.02 − 1.78i)4-s + 1.23·5-s + (0.887 + 1.53i)7-s − 1.84·8-s + (1.07 − 1.86i)10-s + (−0.745 − 1.29i)11-s + (0.0157 − 0.0272i)13-s + 3.10·14-s + (−0.586 + 1.01i)16-s + (0.633 − 1.09i)17-s + (−0.0481 + 0.0833i)19-s + (−1.26 − 2.19i)20-s − 2.60·22-s + (−0.378 + 0.655i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.456 + 0.889i$
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: \((2,\ 603,\ (\ :1/2),\ -0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44058 - 2.35717i\)
\(L(\frac12)\) \(\approx\) \(1.44058 - 2.35717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 8.08i)T \)
good2 \( 1 + (-1.23 + 2.14i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 2.75T + 5T^{2} \)
7 \( 1 + (-2.34 - 4.06i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.47 + 4.28i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0567 + 0.0983i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.61 + 4.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.209 - 0.363i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.81 - 3.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.19 - 5.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.69 + 6.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.63 - 9.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.42 - 7.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + (1.67 + 2.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.66T + 53T^{2} \)
59 \( 1 + 6.66T + 59T^{2} \)
61 \( 1 + (0.347 - 0.601i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (1.81 + 3.14i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.36 - 9.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.17 - 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.140 - 0.243i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.07T + 89T^{2} \)
97 \( 1 + (-9.46 + 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.58374456739144225930120317596, −9.720509271054414492103929283767, −9.036326986257648281540915197735, −8.034985149281749096738846507248, −6.06173512349219540519209783546, −5.43148222013848405567970561410, −4.95902498594905646969267291728, −3.18542318145423039768786316541, −2.48980138563446301179975462967, −1.47311006090642491810496445509, 1.90407883718964159636396143037, 3.87446542797634161852652919483, 4.68141811380407086260304437190, 5.47407701684421727317394474193, 6.39547955481668395269230361540, 7.35651536876252088129875497776, 7.77892704760956988087576488113, 8.902806360383514410764604840364, 10.31341309458413378708881642900, 10.48742655195443515329552174148

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