Properties

Label 2-605-55.4-c1-0-44
Degree $2$
Conductor $605$
Sign $-0.457 - 0.889i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (1.53 − 2.10i)2-s + (−2.03 − 0.661i)3-s + (−1.48 − 4.56i)4-s + (1.27 − 1.83i)5-s + (−4.51 + 3.27i)6-s + (−0.940 + 0.305i)7-s + (−6.94 − 2.25i)8-s + (1.27 + 0.927i)9-s + (−1.92 − 5.50i)10-s + 10.2i·12-s + (−1.80 + 2.48i)13-s + (−0.796 + 2.45i)14-s + (−3.80 + 2.89i)15-s + (−7.63 + 5.55i)16-s + (0.951 + 1.30i)17-s + (3.91 − 1.27i)18-s + ⋯
L(s)  = 1  + (1.08 − 1.49i)2-s + (−1.17 − 0.381i)3-s + (−0.741 − 2.28i)4-s + (0.569 − 0.821i)5-s + (−1.84 + 1.33i)6-s + (−0.355 + 0.115i)7-s + (−2.45 − 0.797i)8-s + (0.425 + 0.309i)9-s + (−0.608 − 1.74i)10-s + 2.96i·12-s + (−0.501 + 0.690i)13-s + (−0.212 + 0.655i)14-s + (−0.983 + 0.748i)15-s + (−1.90 + 1.38i)16-s + (0.230 + 0.317i)17-s + (0.922 − 0.299i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.457 - 0.889i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.457 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713875 + 1.17047i\)
\(L(\frac12)\) \(\approx\) \(0.713875 + 1.17047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.27 + 1.83i)T \)
11 \( 1 \)
good2 \( 1 + (-1.53 + 2.10i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.03 + 0.661i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.940 - 0.305i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.80 - 2.48i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.951 - 1.30i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.03 + 6.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 + (-2.25 - 6.95i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.13 - 1.55i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.66 - 0.866i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.428 - 1.31i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (2.03 + 0.661i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.39 + 1.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.78 + 11.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.810 - 0.589i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + (-0.197 + 0.143i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-12.9 + 4.22i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.22 + 4.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.50 + 4.81i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-7.48 + 10.3i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.43563690751741201864718585095, −9.572753821315917652236498534398, −8.785909759297105550732149046712, −6.78155607520494823840010985562, −6.07960494401696225761008922159, −4.97671525636210777080194554171, −4.76083458437538813313846702036, −3.13666289274239256670235222885, −1.84627972668840476914828817982, −0.61811030556046090152472655153, 2.99352367536910742677900222053, 4.07572483389717116584475998719, 5.29514440749111412157852202176, 5.74161612016958479687397968420, 6.41416378030385644552741540553, 7.33372025851218060760767683874, 8.090808226771023334987072482685, 9.687365407878677267018066593360, 10.28571386173279761103437657552, 11.51196841773825174641131496925

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