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Properties

Label 2-605-55.18-c1-0-25
Degree $2$
Conductor $605$
Sign $-0.628 + 0.778i$
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  + (−0.349 − 2.20i)2-s + (−1.26 + 0.642i)3-s + (−2.85 + 0.927i)4-s + (2.20 − 0.366i)5-s + (1.85 + 2.55i)6-s + (1.01 + 1.99i)8-s + (−0.587 + 0.809i)9-s + (−1.58 − 4.74i)10-s + (3.00 − 3.00i)12-s + (4.41 − 0.699i)13-s + (−2.54 + 1.87i)15-s + (−0.809 + 0.587i)16-s + (4.41 + 0.699i)17-s + (1.99 + 1.01i)18-s + (1.95 − 6.01i)19-s + (−5.95 + 3.09i)20-s + ⋯
L(s)  = 1  + (−0.247 − 1.56i)2-s + (−0.727 + 0.370i)3-s + (−1.42 + 0.463i)4-s + (0.986 − 0.163i)5-s + (0.758 + 1.04i)6-s + (0.358 + 0.704i)8-s + (−0.195 + 0.269i)9-s + (−0.499 − 1.50i)10-s + (0.866 − 0.866i)12-s + (1.22 − 0.194i)13-s + (−0.656 + 0.484i)15-s + (−0.202 + 0.146i)16-s + (1.07 + 0.169i)17-s + (0.469 + 0.239i)18-s + (0.448 − 1.37i)19-s + (−1.33 + 0.691i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.628 + 0.778i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.628 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482329 - 1.00941i\)
\(L(\frac12)\) \(\approx\) \(0.482329 - 1.00941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.20 + 0.366i)T \)
11 \( 1 \)
good2 \( 1 + (0.349 + 2.20i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (1.26 - 0.642i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (-4.41 + 0.699i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-4.41 - 0.699i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-1.95 + 6.01i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (1.95 + 6.01i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.78 - 1.92i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (6.01 + 1.95i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (1.92 + 3.78i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.221 - 1.39i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (5.70 - 1.85i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.71 + 5.11i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + (-6.47 + 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.98 - 2.03i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-5.11 - 3.71i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.39 + 8.83i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-9.77 + 1.54i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.52600246267638925448491646477, −9.762604805652809859715700716673, −9.081584485421814707285243989598, −8.075542861624712611632629382259, −6.41227895224699576854101454511, −5.56593908054509731028918715638, −4.59944241831726348849285064911, −3.33857747566075253785420218757, −2.20425231189419072962805247075, −0.868934800649850759624725034595, 1.34783378167030411417233515428, 3.46638189818756018669056839516, 5.24213408417583334175143227781, 5.77769180976558075720609862542, 6.35756358941609131679925383670, 7.16866028343447014979411068866, 8.164742996682422880373104552123, 9.039911033876166667408842421528, 9.830844687531610104619862138030, 10.85298211164579456534450379304

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