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Properties

Label 2-605-55.43-c1-0-12
Degree $2$
Conductor $605$
Sign $-0.504 - 0.863i$
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  + (1.05 + 1.05i)2-s + (1.53 + 1.53i)3-s + 0.238i·4-s + (−1.92 + 1.13i)5-s + 3.25i·6-s + (1.66 + 1.66i)7-s + (1.86 − 1.86i)8-s + 1.73i·9-s + (−3.23 − 0.835i)10-s + (−0.366 + 0.366i)12-s + (−4.40 + 4.40i)13-s + 3.52i·14-s + (−4.71 − 1.21i)15-s + 4.41·16-s + (3.57 + 3.57i)17-s + (−1.83 + 1.83i)18-s + ⋯
L(s)  = 1  + (0.748 + 0.748i)2-s + (0.888 + 0.888i)3-s + 0.119i·4-s + (−0.861 + 0.508i)5-s + 1.32i·6-s + (0.629 + 0.629i)7-s + (0.658 − 0.658i)8-s + 0.578i·9-s + (−1.02 − 0.264i)10-s + (−0.105 + 0.105i)12-s + (−1.22 + 1.22i)13-s + 0.941i·14-s + (−1.21 − 0.313i)15-s + 1.10·16-s + (0.867 + 0.867i)17-s + (−0.432 + 0.432i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.504 - 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31423 + 2.29034i\)
\(L(\frac12)\) \(\approx\) \(1.31423 + 2.29034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.92 - 1.13i)T \)
11 \( 1 \)
good2 \( 1 + (-1.05 - 1.05i)T + 2iT^{2} \)
3 \( 1 + (-1.53 - 1.53i)T + 3iT^{2} \)
7 \( 1 + (-1.66 - 1.66i)T + 7iT^{2} \)
13 \( 1 + (4.40 - 4.40i)T - 13iT^{2} \)
17 \( 1 + (-3.57 - 3.57i)T + 17iT^{2} \)
19 \( 1 - 0.201T + 19T^{2} \)
23 \( 1 + (2.77 + 2.77i)T + 23iT^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 + 6.45T + 31T^{2} \)
37 \( 1 + (-6.07 + 6.07i)T - 37iT^{2} \)
41 \( 1 - 2.91iT - 41T^{2} \)
43 \( 1 + (-2.16 + 2.16i)T - 43iT^{2} \)
47 \( 1 + (-6.27 + 6.27i)T - 47iT^{2} \)
53 \( 1 + (5.53 + 5.53i)T + 53iT^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + 3.92iT - 61T^{2} \)
67 \( 1 + (0.637 - 0.637i)T - 67iT^{2} \)
71 \( 1 - 1.45T + 71T^{2} \)
73 \( 1 + (-6.80 + 6.80i)T - 73iT^{2} \)
79 \( 1 - 8.04T + 79T^{2} \)
83 \( 1 + (1.70 - 1.70i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (2.07 - 2.07i)T - 97iT^{2} \)
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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

−10.81842420158042804512937721103, −10.00044423217236902834053089931, −9.157951705771232707773667473684, −8.159201635745594507747837690840, −7.42228379904946280025295772794, −6.42179156522163666933043940642, −5.23626487127343389015843197625, −4.31630356463045109679748509708, −3.73002239524093244088534244052, −2.29864745658279660247261982928, 1.18826045554777773085177813305, 2.59069923722348786751670789173, 3.41240861545598177823432812031, 4.55554481170886910193943850127, 5.33126509913540303083652742646, 7.38100611600129239737886352045, 7.69587781638238954504240493764, 8.181725532463111965699476783699, 9.495418129966526711345111389647, 10.66277683968594605968380311397

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