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Properties

Label 2-605-55.18-c1-0-6
Degree $2$
Conductor $605$
Sign $-0.628 + 0.778i$
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  + (0.349 + 2.20i)2-s + (−1.26 + 0.642i)3-s + (−2.85 + 0.927i)4-s + (2.20 − 0.366i)5-s + (−1.85 − 2.55i)6-s + (−1.01 − 1.99i)8-s + (−0.587 + 0.809i)9-s + (1.58 + 4.74i)10-s + (3.00 − 3.00i)12-s + (−4.41 + 0.699i)13-s + (−2.54 + 1.87i)15-s + (−0.809 + 0.587i)16-s + (−4.41 − 0.699i)17-s + (−1.99 − 1.01i)18-s + (−1.95 + 6.01i)19-s + (−5.95 + 3.09i)20-s + ⋯
L(s)  = 1  + (0.247 + 1.56i)2-s + (−0.727 + 0.370i)3-s + (−1.42 + 0.463i)4-s + (0.986 − 0.163i)5-s + (−0.758 − 1.04i)6-s + (−0.358 − 0.704i)8-s + (−0.195 + 0.269i)9-s + (0.499 + 1.50i)10-s + (0.866 − 0.866i)12-s + (−1.22 + 0.194i)13-s + (−0.656 + 0.484i)15-s + (−0.202 + 0.146i)16-s + (−1.07 − 0.169i)17-s + (−0.469 − 0.239i)18-s + (−0.448 + 1.37i)19-s + (−1.33 + 0.691i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.628 + 0.778i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.628 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405618 - 0.848878i\)
\(L(\frac12)\) \(\approx\) \(0.405618 - 0.848878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.20 + 0.366i)T \)
11 \( 1 \)
good2 \( 1 + (-0.349 - 2.20i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (1.26 - 0.642i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (4.41 - 0.699i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (4.41 + 0.699i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.95 - 6.01i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (-1.95 - 6.01i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.78 - 1.92i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-6.01 - 1.95i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (1.92 + 3.78i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.221 - 1.39i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (5.70 - 1.85i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.71 - 5.11i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + (-6.47 + 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.98 + 2.03i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (5.11 + 3.71i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.39 - 8.83i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-9.77 + 1.54i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.03123577351796664976946708150, −10.23183869956558133436592133479, −9.315378670777033210041275948700, −8.440847743584976785907157273340, −7.45750028527755110145875512077, −6.46484028899394460149247961094, −5.86239692107538045936573345869, −5.03946958230128504326206239585, −4.40277114423705436529110060647, −2.27942989526277649049319842440, 0.49654949681548063256370216568, 2.07845120106887990414662762857, 2.81836615160303873764005431408, 4.36202615126042498247175804392, 5.25739039116070913365979211712, 6.32957357324997518612027724993, 7.16761093626725325367720217510, 8.870880966509569672534348885370, 9.542513814632773735022065993592, 10.31290567948970536721611297330

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