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Properties

Label 2-414-9.4-c1-0-18
Degree $2$
Conductor $414$
Sign $-0.522 + 0.852i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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 + (−0.278 − 1.70i)3-s + (−0.499 + 0.866i)4-s + (−1.69 + 2.93i)5-s + (1.34 − 1.09i)6-s + (−1.74 − 3.02i)7-s − 0.999·8-s + (−2.84 + 0.950i)9-s − 3.39·10-s + (−2.39 − 4.15i)11-s + (1.61 + 0.613i)12-s + (2.56 − 4.44i)13-s + (1.74 − 3.02i)14-s + (5.49 + 2.08i)15-s + (−0.5 − 0.866i)16-s − 6.35·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.160 − 0.987i)3-s + (−0.249 + 0.433i)4-s + (−0.758 + 1.31i)5-s + (0.547 − 0.447i)6-s + (−0.659 − 1.14i)7-s − 0.353·8-s + (−0.948 + 0.316i)9-s − 1.07·10-s + (−0.722 − 1.25i)11-s + (0.467 + 0.177i)12-s + (0.711 − 1.23i)13-s + (0.466 − 0.808i)14-s + (1.41 + 0.538i)15-s + (−0.125 − 0.216i)16-s − 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ -0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238546 - 0.426156i\)
\(L(\frac12)\) \(\approx\) \(0.238546 - 0.426156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.278 + 1.70i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1.69 - 2.93i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.74 + 3.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.39 + 4.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.56 + 4.44i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.35T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.88T + 37T^{2} \)
41 \( 1 + (2.74 - 4.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.387 - 0.671i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.55 + 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.41T + 53T^{2} \)
59 \( 1 + (3.08 - 5.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.632 + 1.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.74 - 3.02i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.4T + 71T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 + (3.43 + 5.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.29 + 5.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + (-0.770 - 1.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.89679759105948036813804271151, −10.46420372203698691423215317166, −8.572107124311720134603787946100, −7.85501566443783371880314455304, −6.99211522883328846237932557502, −6.52876290176719061879468876844, −5.42008632619984990052884154967, −3.57808359849548183076835507827, −3.02358258991302223239838101484, −0.27213714686340533917418810016, 2.23221189192443564046680540563, 3.77957005975174960343429160679, 4.63969137781937375696555770447, 5.28480895255464483235407228308, 6.57234724508188059029890681318, 8.319805567048139071737711904956, 9.141598667124666765472351835949, 9.488089350655539122078315001981, 10.74695532911032258539851407341, 11.71742909360755349095779912504

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