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Properties

Label 2-456-456.11-c1-0-31
Degree $2$
Conductor $456$
Sign $0.894 - 0.446i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (1.39 − 0.209i)2-s + (−1.72 − 0.203i)3-s + (1.91 − 0.585i)4-s + (−1.25 + 2.16i)5-s + (−2.44 + 0.0754i)6-s − 0.788i·7-s + (2.55 − 1.21i)8-s + (2.91 + 0.700i)9-s + (−1.29 + 3.29i)10-s + 5.60i·11-s + (−3.40 + 0.618i)12-s + (5.46 − 3.15i)13-s + (−0.165 − 1.10i)14-s + (2.59 − 3.47i)15-s + (3.31 − 2.24i)16-s + (2.43 + 1.40i)17-s + ⋯
L(s)  = 1  + (0.988 − 0.148i)2-s + (−0.993 − 0.117i)3-s + (0.956 − 0.292i)4-s + (−0.559 + 0.968i)5-s + (−0.999 + 0.0308i)6-s − 0.298i·7-s + (0.902 − 0.431i)8-s + (0.972 + 0.233i)9-s + (−0.409 + 1.04i)10-s + 1.68i·11-s + (−0.983 + 0.178i)12-s + (1.51 − 0.874i)13-s + (−0.0441 − 0.294i)14-s + (0.669 − 0.896i)15-s + (0.828 − 0.560i)16-s + (0.590 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82605 + 0.430031i\)
\(L(\frac12)\) \(\approx\) \(1.82605 + 0.430031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.209i)T \)
3 \( 1 + (1.72 + 0.203i)T \)
19 \( 1 + (2.46 - 3.59i)T \)
good5 \( 1 + (1.25 - 2.16i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.788iT - 7T^{2} \)
11 \( 1 - 5.60iT - 11T^{2} \)
13 \( 1 + (-5.46 + 3.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.43 - 1.40i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.30 - 3.98i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.39 + 2.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.92iT - 31T^{2} \)
37 \( 1 - 5.33iT - 37T^{2} \)
41 \( 1 + (6.96 + 4.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.39 - 2.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.96 + 5.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.63 + 2.82i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.16 + 2.40i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.466 + 0.269i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 5.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.217 + 0.377i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.40 - 5.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (12.4 + 7.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.1iT - 83T^{2} \)
89 \( 1 + (-0.0553 + 0.0319i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.60 + 11.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.34783886096662114901426479269, −10.40726374932045391103484405904, −10.06222051942014868450725946133, −7.86812468941879483687631761765, −7.19846689954001013479332102937, −6.34619014915555868291531002253, −5.49326052941774887955031912800, −4.25268121698071548245067163541, −3.45421152875583322736480040230, −1.67250774939096330208944819789, 1.13318998208881698657181354893, 3.35631248034036715070452945899, 4.37170890467496243344466278882, 5.25036405198714435803240547741, 6.11029137812020980704711204852, 6.88438539729053604490786952399, 8.348548369841235623543453787183, 8.932662545411425344964494072414, 10.75249355690663577426200371968, 11.14077613693554205175637624335

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