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Properties

Label 2-603-67.64-c1-0-17
Degree $2$
Conductor $603$
Sign $0.474 + 0.880i$
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  + (0.459 + 0.134i)2-s + (−1.48 − 0.957i)4-s + (0.345 + 0.755i)5-s + (1.04 + 0.306i)7-s + (−1.18 − 1.36i)8-s + (0.0566 + 0.393i)10-s + (−1.19 − 2.61i)11-s + (2.87 − 3.31i)13-s + (0.438 + 0.281i)14-s + (1.11 + 2.43i)16-s + (2.76 − 1.77i)17-s + (−3.40 + 1.00i)19-s + (0.209 − 1.45i)20-s + (−0.196 − 1.36i)22-s + (0.912 − 6.34i)23-s + ⋯
L(s)  = 1  + (0.324 + 0.0954i)2-s + (−0.744 − 0.478i)4-s + (0.154 + 0.337i)5-s + (0.394 + 0.115i)7-s + (−0.418 − 0.482i)8-s + (0.0179 + 0.124i)10-s + (−0.360 − 0.788i)11-s + (0.797 − 0.920i)13-s + (0.117 + 0.0752i)14-s + (0.277 + 0.608i)16-s + (0.670 − 0.430i)17-s + (−0.781 + 0.229i)19-s + (0.0468 − 0.325i)20-s + (−0.0418 − 0.290i)22-s + (0.190 − 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ 0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22532 - 0.731433i\)
\(L(\frac12)\) \(\approx\) \(1.22532 - 0.731433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.85 + 2.31i)T \)
good2 \( 1 + (-0.459 - 0.134i)T + (1.68 + 1.08i)T^{2} \)
5 \( 1 + (-0.345 - 0.755i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (-1.04 - 0.306i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (1.19 + 2.61i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-2.87 + 3.31i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-2.76 + 1.77i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (3.40 - 1.00i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.912 + 6.34i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 - 1.97T + 29T^{2} \)
31 \( 1 + (1.76 + 2.03i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 - 8.07T + 37T^{2} \)
41 \( 1 + (-2.28 + 1.46i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (6.51 - 4.18i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.56 + 10.8i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (6.30 + 4.05i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-6.97 - 8.05i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (3.84 - 8.41i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-1.45 - 0.937i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.608 - 1.33i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (10.1 - 11.7i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-3.35 - 7.35i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-2.19 - 15.2i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 - 1.03T + 97T^{2} \)
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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

−10.50245002283793316308983183845, −9.786067855495474898534011641829, −8.508166480408739706467736791716, −8.221258678571908471660608811068, −6.63756001511679545654978161954, −5.84580516861274933860177395973, −5.04700071691323749428260647541, −3.92723431732745209371151891812, −2.74783520993624819418453132830, −0.798764712563969764476399839761, 1.61232979622665992908402488292, 3.25226509064300584966177298576, 4.33850828896076085544327886517, 5.02055970368665032201019256496, 6.13631392051022649619918797939, 7.41812854729268358296289814531, 8.215750294503093296312350935468, 9.093517065484486095005176863753, 9.736846255578706692027654818075, 10.97233986269412535931896939103

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