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Properties

Label 2-605-11.4-c1-0-30
Degree $2$
Conductor $605$
Sign $0.927 + 0.374i$
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  + (1.69 + 1.23i)2-s + (0.591 − 1.81i)3-s + (0.738 + 2.27i)4-s + (0.809 − 0.587i)5-s + (3.24 − 2.35i)6-s + (−0.947 − 2.91i)7-s + (−0.252 + 0.777i)8-s + (−0.533 − 0.387i)9-s + 2.09·10-s + 4.57·12-s + (−2.46 − 1.79i)13-s + (1.98 − 6.11i)14-s + (−0.591 − 1.81i)15-s + (2.48 − 1.80i)16-s + (0.375 − 0.272i)17-s + (−0.427 − 1.31i)18-s + ⋯
L(s)  = 1  + (1.19 + 0.870i)2-s + (0.341 − 1.05i)3-s + (0.369 + 1.13i)4-s + (0.361 − 0.262i)5-s + (1.32 − 0.961i)6-s + (−0.358 − 1.10i)7-s + (−0.0893 + 0.274i)8-s + (−0.177 − 0.129i)9-s + 0.662·10-s + 1.31·12-s + (−0.684 − 0.497i)13-s + (0.530 − 1.63i)14-s + (−0.152 − 0.469i)15-s + (0.620 − 0.450i)16-s + (0.0910 − 0.0661i)17-s + (−0.100 − 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05215 - 0.592835i\)
\(L(\frac12)\) \(\approx\) \(3.05215 - 0.592835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 1.23i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.591 + 1.81i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.947 + 2.91i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.46 + 1.79i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.375 + 0.272i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.43 - 7.50i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 + (-1.15 - 3.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.47 - 6.15i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.569 - 1.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.36 - 4.19i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.31T + 43T^{2} \)
47 \( 1 + (0.920 - 2.83i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.38 + 2.46i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.869 - 2.67i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.63 + 1.18i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + (-5.27 + 3.83i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.05 + 9.39i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.35 - 6.79i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.21 - 5.24i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (12.4 + 9.02i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.41021224117260449219577153745, −9.921167309049177768735438000034, −8.246584226677858636054181150207, −7.70714090355669859366367805049, −6.80511058825491468309817588684, −6.28486500776782573605461294604, −5.12152355226017497936980871097, −4.16619877207884490193513624495, −3.00241336043977762932842466001, −1.32575355927140503506158525281, 2.33399294784449517377393685713, 2.87961495134505632905460470498, 4.13207447113387783271568348887, 4.79236207261004497702509128074, 5.75789033885341689538517829777, 6.75840450136717518808565371451, 8.402374640849349343920970715079, 9.348032776127155398628986709953, 9.912816624596862016254430260127, 10.82076880610778169163309643748

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