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Properties

Label 2-605-11.3-c1-0-20
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: $\chi_{605} (366, \cdot )$
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

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

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