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Properties

Label 2-605-55.2-c1-0-19
Degree $2$
Conductor $605$
Sign $-0.585 + 0.810i$
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  + (−1.10 − 2.16i)2-s + (−0.0935 + 0.590i)3-s + (−2.30 + 3.16i)4-s + (1.76 − 1.37i)5-s + (1.38 − 0.449i)6-s + (−2.54 + 0.403i)7-s + (4.60 + 0.729i)8-s + (2.51 + 0.816i)9-s + (−4.92 − 2.31i)10-s + (−1.65 − 1.65i)12-s + (2.50 − 1.27i)13-s + (3.69 + 5.08i)14-s + (0.645 + 1.17i)15-s + (−1.08 − 3.33i)16-s + (−0.841 − 0.428i)17-s + (−1.00 − 6.34i)18-s + ⋯
L(s)  = 1  + (−0.780 − 1.53i)2-s + (−0.0540 + 0.341i)3-s + (−1.15 + 1.58i)4-s + (0.789 − 0.613i)5-s + (0.564 − 0.183i)6-s + (−0.963 + 0.152i)7-s + (1.62 + 0.257i)8-s + (0.837 + 0.272i)9-s + (−1.55 − 0.731i)10-s + (−0.478 − 0.478i)12-s + (0.694 − 0.353i)13-s + (0.986 + 1.35i)14-s + (0.166 + 0.302i)15-s + (−0.271 − 0.834i)16-s + (−0.204 − 0.103i)17-s + (−0.237 − 1.49i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448663 - 0.877020i\)
\(L(\frac12)\) \(\approx\) \(0.448663 - 0.877020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.76 + 1.37i)T \)
11 \( 1 \)
good2 \( 1 + (1.10 + 2.16i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (0.0935 - 0.590i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (2.54 - 0.403i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.50 + 1.27i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.841 + 0.428i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-3.35 + 2.43i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.104 + 0.104i)T - 23iT^{2} \)
29 \( 1 + (-5.61 - 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.80 + 8.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.18 + 7.50i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (2.02 + 2.78i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (3.91 + 3.91i)T + 43iT^{2} \)
47 \( 1 + (-1.04 - 0.165i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (1.87 + 3.67i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (3.80 - 5.24i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.05 + 2.94i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.94 - 2.94i)T + 67iT^{2} \)
71 \( 1 + (-0.391 - 1.20i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.277 - 1.74i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (1.34 - 4.12i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.76 - 7.38i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 4.23iT - 89T^{2} \)
97 \( 1 + (-12.8 + 6.54i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.20503153345888611502440689602, −9.665374299243899120899913063580, −9.085165262851062451551529476607, −8.222134565044370666666551936728, −6.84913578669115834141645419917, −5.57926917509287900613771593418, −4.36566303587195473940275046497, −3.30003996258343232523553086506, −2.16727798295544652852023246220, −0.852980015809675725074841137888, 1.30447691608866061953871766682, 3.27105394509783947226553400701, 4.88452595679287931744786027279, 6.23113040314833723043571391269, 6.44665119098860870293911203789, 7.17440303868341915743527288214, 8.181833786980345046814188840841, 9.152879817283297627603305819864, 9.977075740751362135538777188367, 10.26385469348763716047689053338

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