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Properties

Label 2-605-55.4-c1-0-26
Degree $2$
Conductor $605$
Sign $0.242 + 0.970i$
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  + (−1.53 + 2.10i)2-s + (−2.03 − 0.661i)3-s + (−1.48 − 4.56i)4-s + (1.27 − 1.83i)5-s + (4.51 − 3.27i)6-s + (0.940 − 0.305i)7-s + (6.94 + 2.25i)8-s + (1.27 + 0.927i)9-s + (1.92 + 5.50i)10-s + 10.2i·12-s + (1.80 − 2.48i)13-s + (−0.796 + 2.45i)14-s + (−3.80 + 2.89i)15-s + (−7.63 + 5.55i)16-s + (−0.951 − 1.30i)17-s + (−3.91 + 1.27i)18-s + ⋯
L(s)  = 1  + (−1.08 + 1.49i)2-s + (−1.17 − 0.381i)3-s + (−0.741 − 2.28i)4-s + (0.569 − 0.821i)5-s + (1.84 − 1.33i)6-s + (0.355 − 0.115i)7-s + (2.45 + 0.797i)8-s + (0.425 + 0.309i)9-s + (0.608 + 1.74i)10-s + 2.96i·12-s + (0.501 − 0.690i)13-s + (−0.212 + 0.655i)14-s + (−0.983 + 0.748i)15-s + (−1.90 + 1.38i)16-s + (−0.230 − 0.317i)17-s + (−0.922 + 0.299i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.276509 - 0.215974i\)
\(L(\frac12)\) \(\approx\) \(0.276509 - 0.215974i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.27 + 1.83i)T \)
11 \( 1 \)
good2 \( 1 + (1.53 - 2.10i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.03 + 0.661i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.940 + 0.305i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.80 + 2.48i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.951 + 1.30i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.03 - 6.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 + (2.25 + 6.95i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.13 - 1.55i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.66 - 0.866i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.428 + 1.31i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + (2.03 + 0.661i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.39 + 1.92i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.78 + 11.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.810 + 0.589i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + (-0.197 + 0.143i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (12.9 - 4.22i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.22 - 4.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.50 - 4.81i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-7.48 + 10.3i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.25987884098861623594509376709, −9.467303732762763672579998099394, −8.383005053276025075494466126935, −7.984403082725007229172464595007, −6.69013337915892561146960986908, −6.07180291591138252343446663918, −5.47664124275461050109797878193, −4.57636027728251221790474167228, −1.52378504783128994037091532946, −0.35134021688521361690782925150, 1.52611619965912607845908753789, 2.71959011772098370166284571877, 3.97181366230112736509774718219, 5.16991325889313270485222545670, 6.41409591277348513991065761214, 7.39246285036723836949361834346, 8.729420928877882182254838573392, 9.335328348224193070980652912146, 10.36901365796288790711921577569, 10.76631159738576266068158294929

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