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Properties

Label 2-574-7.2-c1-0-14
Degree $2$
Conductor $574$
Sign $0.968 - 0.250i$
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 + (1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 3·6-s + (−0.5 − 2.59i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (−0.499 − 0.866i)10-s + (2.5 + 4.33i)11-s + (1.50 − 2.59i)12-s + 6·13-s + (−2.5 − 0.866i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 1.22·6-s + (−0.188 − 0.981i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (−0.158 − 0.273i)10-s + (0.753 + 1.30i)11-s + (0.433 − 0.749i)12-s + 1.66·13-s + (−0.668 − 0.231i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(574\)    =    \(2 \cdot 7 \cdot 41\)
Sign: $0.968 - 0.250i$
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: \((2,\ 574,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37577 + 0.302753i\)
\(L(\frac12)\) \(\approx\) \(2.37577 + 0.302753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 2.59i)T \)
41 \( 1 + T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.78621845620538618356233269352, −9.784771335468472194050942062808, −9.329248559587861696407246346810, −8.565982591799042366007769641897, −7.28887872269846052056802382656, −5.88128504112401128923391485739, −4.60346180801016421402366394319, −4.05677065618308965971246833264, −3.31995324715127549529818601653, −1.69392725409907675149484976380, 1.42991334075189403462843659005, 2.88952621977859999751216783963, 3.61506662051115475649010061014, 5.73998904769762330371571931397, 6.21254414506662038972806461962, 6.97297775455647688475100763505, 8.044062185996951581940765820644, 8.797241951992277716484153274471, 9.121586304814477802799895598805, 11.02264732983570531156208468251

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