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Properties

Label 2-605-55.14-c1-0-27
Degree $2$
Conductor $605$
Sign $-0.904 + 0.426i$
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  + (−1.13 − 1.56i)2-s + (−0.492 + 0.159i)3-s + (−0.535 + 1.64i)4-s + (2.23 − 0.126i)5-s + (0.809 + 0.587i)6-s + (0.852 + 0.277i)7-s + (−0.492 + 0.159i)8-s + (−2.21 + 1.60i)9-s + (−2.73 − 3.34i)10-s − 0.896i·12-s + (−2.49 − 3.43i)13-s + (−0.535 − 1.64i)14-s + (−1.07 + 0.419i)15-s + (3.61 + 2.62i)16-s + (0.608 − 0.837i)17-s + (5.01 + 1.63i)18-s + ⋯
L(s)  = 1  + (−0.802 − 1.10i)2-s + (−0.284 + 0.0923i)3-s + (−0.267 + 0.823i)4-s + (0.998 − 0.0563i)5-s + (0.330 + 0.239i)6-s + (0.322 + 0.104i)7-s + (−0.174 + 0.0565i)8-s + (−0.736 + 0.535i)9-s + (−0.863 − 1.05i)10-s − 0.258i·12-s + (−0.691 − 0.951i)13-s + (−0.143 − 0.440i)14-s + (−0.278 + 0.108i)15-s + (0.902 + 0.655i)16-s + (0.147 − 0.203i)17-s + (1.18 + 0.384i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.904 + 0.426i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.904 + 0.426i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166412 - 0.744013i\)
\(L(\frac12)\) \(\approx\) \(0.166412 - 0.744013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 + 0.126i)T \)
11 \( 1 \)
good2 \( 1 + (1.13 + 1.56i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.492 - 0.159i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-0.852 - 0.277i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.49 + 3.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.608 + 0.837i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + (-2.14 + 6.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.26 + 3.09i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.36 + 2.06i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.535 + 1.64i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (-3.90 + 1.26i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.03 + 9.68i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.46 + 4.50i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.68 - 4.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 + (1.77 + 1.29i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.65 - 1.51i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.42 - 3.21i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.81 - 8.00i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + (-0.385 - 0.530i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.22744896499811190976812766967, −9.745743781417069499620766650632, −8.681109753396475262530828680941, −8.127650452661292259190894845886, −6.59290179083389461903993436492, −5.59351180945551312727835134753, −4.74080706219987177228122605960, −2.77636511324926121486000730815, −2.29175320930009489186819091427, −0.56741193576890193644737118818, 1.58138099019158889071061216526, 3.28773844061893241557436607353, 5.04821301220613177748251579717, 5.89364170100146422056369447392, 6.57616472844571007389316139790, 7.38789823919482603983755703501, 8.474697220777082712493015669248, 9.114510410271207943554771850681, 9.875384463530931305517098744004, 10.72208772298441555146113369189

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