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Properties

Label 2-605-55.9-c1-0-0
Degree $2$
Conductor $605$
Sign $-0.615 - 0.788i$
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  + (0.492 + 0.159i)2-s + (−1.13 − 1.56i)3-s + (−1.40 − 1.01i)4-s + (1.88 + 1.21i)5-s + (−0.309 − 0.951i)6-s + (−1.96 + 2.70i)7-s + (−1.13 − 1.56i)8-s + (−0.226 + 0.696i)9-s + (0.732 + 0.896i)10-s + 3.34i·12-s + (−4.03 − 1.31i)13-s + (−1.40 + 1.01i)14-s + (−0.243 − 4.31i)15-s + (0.761 + 2.34i)16-s + (−3.67 + 1.19i)17-s + (−0.222 + 0.306i)18-s + ⋯
L(s)  = 1  + (0.348 + 0.113i)2-s + (−0.655 − 0.902i)3-s + (−0.700 − 0.509i)4-s + (0.840 + 0.541i)5-s + (−0.126 − 0.388i)6-s + (−0.743 + 1.02i)7-s + (−0.401 − 0.552i)8-s + (−0.0754 + 0.232i)9-s + (0.231 + 0.283i)10-s + 0.965i·12-s + (−1.11 − 0.363i)13-s + (−0.374 + 0.272i)14-s + (−0.0628 − 1.11i)15-s + (0.190 + 0.585i)16-s + (−0.891 + 0.289i)17-s + (−0.0524 + 0.0722i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0949616 + 0.194578i\)
\(L(\frac12)\) \(\approx\) \(0.0949616 + 0.194578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.88 - 1.21i)T \)
11 \( 1 \)
good2 \( 1 + (-0.492 - 0.159i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.13 + 1.56i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.96 - 2.70i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.03 + 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.67 - 1.19i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.39 - 2.46i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (-5.60 - 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.69 - 8.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 + 1.45i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.40 - 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.45iT - 43T^{2} \)
47 \( 1 + (6.73 + 9.26i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.06 + 0.671i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.02 + 0.745i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.62 + 11.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (2.53 + 7.79i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.87 + 3.96i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.452 - 1.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.41 - 3.05i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + (8.69 + 2.82i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.87218284400876682275676962229, −10.07071382808632705430684722560, −9.327893403747614242627422058459, −8.453868424123306625046818450094, −6.80997599875402437302718836942, −6.45514539512421991723173878892, −5.67219888896954753922553720866, −4.82339287981952342932556128002, −3.10695362072873840043208707648, −1.82659809853360574016425756230, 0.11115922550083756755463854321, 2.54794128241362516756212534666, 4.14625677075281126834964682737, 4.53064576342567785327510788729, 5.44640613466910827386305112365, 6.53558254695910591362797188261, 7.67169463138938737754116339939, 8.952791691938934053099258616130, 9.660599666793706160201821033185, 10.12786470040554364429669139780

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