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Properties

Label 2-603-67.9-c1-0-15
Degree $2$
Conductor $603$
Sign $-0.406 + 0.913i$
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  + (−1.34 − 0.861i)2-s + (0.225 + 0.492i)4-s + (0.857 − 0.989i)5-s + (0.313 + 0.201i)7-s + (−0.330 + 2.30i)8-s + (−2.00 + 0.588i)10-s + (0.986 − 1.13i)11-s + (0.00388 + 0.0270i)13-s + (−0.246 − 0.540i)14-s + (3.13 − 3.62i)16-s + (2.04 − 4.47i)17-s + (3.71 − 2.38i)19-s + (0.681 + 0.199i)20-s + (−2.30 + 0.676i)22-s + (−2.47 − 0.727i)23-s + ⋯
L(s)  = 1  + (−0.948 − 0.609i)2-s + (0.112 + 0.246i)4-s + (0.383 − 0.442i)5-s + (0.118 + 0.0761i)7-s + (−0.116 + 0.813i)8-s + (−0.633 + 0.186i)10-s + (0.297 − 0.343i)11-s + (0.00107 + 0.00749i)13-s + (−0.0659 − 0.144i)14-s + (0.784 − 0.905i)16-s + (0.495 − 1.08i)17-s + (0.852 − 0.547i)19-s + (0.152 + 0.0447i)20-s + (−0.491 + 0.144i)22-s + (−0.516 − 0.151i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.406 + 0.913i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.476665 - 0.734058i\)
\(L(\frac12)\) \(\approx\) \(0.476665 - 0.734058i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.71 + 6.69i)T \)
good2 \( 1 + (1.34 + 0.861i)T + (0.830 + 1.81i)T^{2} \)
5 \( 1 + (-0.857 + 0.989i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-0.313 - 0.201i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.986 + 1.13i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.00388 - 0.0270i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-2.04 + 4.47i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-3.71 + 2.38i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (2.47 + 0.727i)T + (19.3 + 12.4i)T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 + (0.0140 - 0.0979i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + 3.22T + 37T^{2} \)
41 \( 1 + (-2.97 + 6.51i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-3.97 + 8.71i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (11.0 + 3.24i)T + (39.5 + 25.4i)T^{2} \)
53 \( 1 + (-4.53 - 9.93i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-1.82 + 12.7i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (7.39 + 8.54i)T + (-8.68 + 60.3i)T^{2} \)
71 \( 1 + (-5.49 - 12.0i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-3.77 - 4.35i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.437 - 3.04i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (3.04 - 3.51i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (3.19 - 0.938i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−10.21232203549126780202316683064, −9.464478474730339165336885518323, −8.950057176179929734682698494121, −8.032174376721227091298488294476, −7.01621417461345646742117750151, −5.62004698528494142731642314799, −4.98116240372652927832176750405, −3.31242985769767016146511747823, −2.01525937042674858074279799792, −0.74103690231089633391414768100, 1.44510642243594983477890508102, 3.18374799624901267342019444582, 4.35916210913977885861176746532, 5.90481306890542454153551254113, 6.54027312678975043561684526914, 7.62245757187523013217271150543, 8.142706543121453535720553818678, 9.198167096674305518734674217184, 9.955500955277317703923766005127, 10.49210603973364103233223393356

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