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Properties

Label 2-603-67.24-c1-0-16
Degree $2$
Conductor $603$
Sign $0.870 - 0.492i$
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  + (−0.915 + 2.00i)2-s + (−1.87 − 2.15i)4-s + (0.0405 − 0.281i)5-s + (0.226 − 0.496i)7-s + (1.80 − 0.531i)8-s + (0.527 + 0.339i)10-s + (0.508 − 3.53i)11-s + (0.942 + 0.276i)13-s + (0.788 + 0.909i)14-s + (0.221 − 1.53i)16-s + (−0.862 + 0.995i)17-s + (−3.05 − 6.67i)19-s + (−0.683 + 0.439i)20-s + (6.62 + 4.26i)22-s + (6.22 − 4.00i)23-s + ⋯
L(s)  = 1  + (−0.647 + 1.41i)2-s + (−0.935 − 1.07i)4-s + (0.0181 − 0.125i)5-s + (0.0857 − 0.187i)7-s + (0.639 − 0.187i)8-s + (0.166 + 0.107i)10-s + (0.153 − 1.06i)11-s + (0.261 + 0.0767i)13-s + (0.210 + 0.243i)14-s + (0.0553 − 0.384i)16-s + (−0.209 + 0.241i)17-s + (−0.699 − 1.53i)19-s + (−0.152 + 0.0982i)20-s + (1.41 + 0.908i)22-s + (1.29 − 0.834i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ 0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884972 + 0.233192i\)
\(L(\frac12)\) \(\approx\) \(0.884972 + 0.233192i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 8.09i)T \)
good2 \( 1 + (0.915 - 2.00i)T + (-1.30 - 1.51i)T^{2} \)
5 \( 1 + (-0.0405 + 0.281i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.226 + 0.496i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (-0.508 + 3.53i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.942 - 0.276i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.862 - 0.995i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (3.05 + 6.67i)T + (-12.4 + 14.3i)T^{2} \)
23 \( 1 + (-6.22 + 4.00i)T + (9.55 - 20.9i)T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
31 \( 1 + (5.85 - 1.71i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 - 5.01T + 37T^{2} \)
41 \( 1 + (0.877 - 1.01i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (-5.80 + 6.69i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (4.50 - 2.89i)T + (19.5 - 42.7i)T^{2} \)
53 \( 1 + (4.58 + 5.28i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-0.837 + 0.245i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (1.86 + 13.0i)T + (-58.5 + 17.1i)T^{2} \)
71 \( 1 + (-2.89 - 3.34i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.738 - 5.13i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-7.48 - 2.19i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-0.929 + 6.46i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (12.8 + 8.24i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−10.75294630385185196754519596298, −9.332442293681963557315751719978, −8.807815882709493434978025767860, −8.191325688290552958010588923391, −7.00068611153234765768778291791, −6.54608303178997476274612462921, −5.48201653287574583474015780983, −4.54571298387157011277542632358, −2.93788994053725545507424728846, −0.71296569238839075747785582248, 1.35228526848397167815632241940, 2.45003648855993839461565855809, 3.58470208251434575435242562049, 4.65568027000932088492085096696, 6.04962421033144257701622862228, 7.24076887225327128573073779367, 8.286160296969559053972357744696, 9.162727359287617861101571452753, 9.804988379425245295050675159001, 10.65922681050430827374967709521

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