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Properties

Label 2-574-7.2-c1-0-13
Degree $2$
Conductor $574$
Sign $0.967 + 0.254i$
Analytic cond. $4.58341$
Root an. cond. $2.14089$
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  + (0.5 − 0.866i)2-s + (0.243 + 0.421i)3-s + (−0.499 − 0.866i)4-s + (−0.308 + 0.533i)5-s + 0.486·6-s + (−0.833 + 2.51i)7-s − 0.999·8-s + (1.38 − 2.39i)9-s + (0.308 + 0.533i)10-s + (0.840 + 1.45i)11-s + (0.243 − 0.421i)12-s + 5.11·13-s + (1.75 + 1.97i)14-s − 0.299·15-s + (−0.5 + 0.866i)16-s + (2.18 + 3.79i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.140 + 0.243i)3-s + (−0.249 − 0.433i)4-s + (−0.137 + 0.238i)5-s + 0.198·6-s + (−0.315 + 0.949i)7-s − 0.353·8-s + (0.460 − 0.797i)9-s + (0.0974 + 0.168i)10-s + (0.253 + 0.438i)11-s + (0.0701 − 0.121i)12-s + 1.41·13-s + (0.469 + 0.528i)14-s − 0.0773·15-s + (−0.125 + 0.216i)16-s + (0.531 + 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $0.967 + 0.254i$
Analytic conductor: \(4.58341\)
Root analytic conductor: \(2.14089\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{574} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 574,\ (\ :1/2),\ 0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85140 - 0.239312i\)
\(L(\frac12)\) \(\approx\) \(1.85140 - 0.239312i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.833 - 2.51i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.243 - 0.421i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.308 - 0.533i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.840 - 1.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.11T + 13T^{2} \)
17 \( 1 + (-2.18 - 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.82 + 6.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.20 - 2.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.00T + 29T^{2} \)
31 \( 1 + (2.01 + 3.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.07 - 3.59i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 9.35T + 43T^{2} \)
47 \( 1 + (1.45 - 2.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.91 - 5.04i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.67 + 9.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.539 + 0.934i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.31 + 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + (-2.29 - 3.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.74 - 8.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.98T + 83T^{2} \)
89 \( 1 + (-1.10 + 1.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−10.82902646514211496569597789442, −9.721649981374376300839782273061, −9.214616945601133631033699728017, −8.306284137828751838584156694382, −6.85405370002284207780175410444, −6.08158478806329970982969196925, −4.98245860132369591276691878538, −3.72884125746976027790974119736, −3.04986023358747457599940284285, −1.42476501515948798118123357317, 1.21267807448722870540098456601, 3.25020626172787345342084753670, 4.12954797860406001955296503360, 5.24546169846766605848725073577, 6.30874674619752336389624749951, 7.17940580490164955962793119190, 8.004635779190170756358808547146, 8.678244069981974123665952505513, 9.986727284650470467621351139375, 10.63824020139544634172806557782

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