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Properties

Label 2-605-55.4-c1-0-31
Degree $2$
Conductor $605$
Sign $0.886 + 0.462i$
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 + (0.570 − 2.16i)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 + (0.626 − 0.973i)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 + (0.254 − 0.966i)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 + (0.161 − 0.251i)15-s + (0.902 − 0.655i)16-s + (0.147 + 0.203i)17-s + (1.18 − 0.384i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 + 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.886 + 0.462i$
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: \((2,\ 605,\ (\ :1/2),\ 0.886 + 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778564 - 0.190692i\)
\(L(\frac12)\) \(\approx\) \(0.778564 - 0.190692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.570 + 2.16i)T \)
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

−10.09337247891534995692668674412, −9.304228003576267657278006768488, −8.782187379451641547370377884022, −8.136902189710474707167549716066, −7.11311139771125455960059117602, −6.25767766519057572792338902541, −5.27227051199204640099139423927, −4.23224751596700724906016726995, −2.52073015871224029696458863175, −0.55696067756809252936230166068, 1.62076957361126438241374947157, 2.76221463289975297741641252044, 3.36558812112010532304366584409, 5.28137121558417968661043571835, 6.13904119703345425847286225039, 7.77652614397535796732383392891, 7.963078036859211986586720575901, 9.368944939786011721099674254266, 9.881091946564956024923816519605, 10.72796829829636492423795756158

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