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Properties

Label 2-603-67.29-c1-0-13
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.535i)2-s + (0.809 + 1.40i)4-s − 3.23·5-s + (−2.11 − 3.66i)7-s − 2.23·8-s + (1.00 − 1.73i)10-s + (2.5 + 4.33i)11-s + (2.11 − 3.66i)13-s + 2.61·14-s + (−0.927 + 1.60i)16-s + (1.11 − 1.93i)17-s + (3.11 − 5.40i)19-s + (−2.61 − 4.53i)20-s − 3.09·22-s + (3.11 − 5.40i)23-s + ⋯
L(s)  = 1  + (−0.218 + 0.378i)2-s + (0.404 + 0.700i)4-s − 1.44·5-s + (−0.800 − 1.38i)7-s − 0.790·8-s + (0.316 − 0.547i)10-s + (0.753 + 1.30i)11-s + (0.587 − 1.01i)13-s + 0.699·14-s + (−0.231 + 0.401i)16-s + (0.271 − 0.469i)17-s + (0.715 − 1.23i)19-s + (−0.585 − 1.01i)20-s − 0.658·22-s + (0.650 − 1.12i)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

\(L(1)\) \(\approx\) \(0.723230 - 0.323094i\)
\(L(\frac12)\) \(\approx\) \(0.723230 - 0.323094i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.535i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.11 + 3.66i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.11 + 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 + 5.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.11 + 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.73 + 8.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.35 - 5.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.736 - 1.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.118 + 0.204i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5.23T + 43T^{2} \)
47 \( 1 + (3.11 + 5.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.70T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + (-0.354 + 0.613i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (6.73 + 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.35 + 9.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.263 + 0.457i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59246090807973342893450992947, −9.632325469279152034063112806940, −8.549728578332323230815585077914, −7.63626306364832701730420609896, −7.13510310342063844884222072762, −6.56627755430118602989923920716, −4.67010780581993244588184156276, −3.78441205164395658498813310338, −3.04464040893017598692316480407, −0.51066235767988318954490241969, 1.40377611171388046596380153876, 3.13627296688868718470029969943, 3.75495922566859999887485232603, 5.55004112606463072525605142211, 6.12862312692736216377773832624, 7.15666789280557139935990879055, 8.447386907755833098184403996763, 9.012285970485267947248988291691, 9.817624236539023443630953622621, 11.10852846410509919882057406730

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