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Properties

Label 2-605-11.3-c1-0-5
Degree $2$
Conductor $605$
Sign $-0.970 - 0.242i$
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  + (0.0756 − 0.0549i)2-s + (0.453 + 1.39i)3-s + (−0.615 + 1.89i)4-s + (−0.809 − 0.587i)5-s + (0.110 + 0.0806i)6-s + (−1.39 + 4.30i)7-s + (0.115 + 0.354i)8-s + (0.686 − 0.498i)9-s − 0.0935·10-s − 2.92·12-s + (−0.924 + 0.671i)13-s + (0.130 + 0.402i)14-s + (0.453 − 1.39i)15-s + (−3.19 − 2.32i)16-s + (2.72 + 1.98i)17-s + (0.0245 − 0.0754i)18-s + ⋯
L(s)  = 1  + (0.0534 − 0.0388i)2-s + (0.261 + 0.805i)3-s + (−0.307 + 0.946i)4-s + (−0.361 − 0.262i)5-s + (0.0452 + 0.0329i)6-s + (−0.528 + 1.62i)7-s + (0.0407 + 0.125i)8-s + (0.228 − 0.166i)9-s − 0.0295·10-s − 0.843·12-s + (−0.256 + 0.186i)13-s + (0.0349 + 0.107i)14-s + (0.117 − 0.360i)15-s + (−0.798 − 0.580i)16-s + (0.661 + 0.480i)17-s + (0.00578 − 0.0177i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128348 + 1.04410i\)
\(L(\frac12)\) \(\approx\) \(0.128348 + 1.04410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.0756 + 0.0549i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.453 - 1.39i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (1.39 - 4.30i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.924 - 0.671i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.72 - 1.98i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.88 + 5.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.45T + 23T^{2} \)
29 \( 1 + (1.02 - 3.15i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.44 - 1.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.460 + 1.41i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.539 - 1.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.263T + 43T^{2} \)
47 \( 1 + (-2.13 - 6.58i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.16 + 0.846i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.18 - 6.72i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.02 - 1.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.516T + 67T^{2} \)
71 \( 1 + (-8.68 - 6.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.75 + 5.40i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.14 - 6.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.62 - 2.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-2.71 + 1.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.15235010711740007699826859212, −9.912602548291557088402675332932, −9.103655686045317812281292584329, −8.755886913741921211842082736228, −7.74673403647742467553068597033, −6.57110383816595264327076088065, −5.34472402735049966140569054857, −4.37860547133476898091053183192, −3.45944109648407894069149475793, −2.49407299857623984485928534609, 0.55790125013480933785255429363, 1.86728627734089396163499262235, 3.62922732704221439670024846826, 4.48866194315912875393263561267, 5.87191046618935497641358933987, 6.77636041840381069249068554467, 7.51125026574197326846755754751, 8.193046716361988840884648652970, 9.762163050535452650683793737265, 10.14831865966420228049733827407

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