Properties

Label 2-460-23.9-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.978 - 0.206i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (−0.671 − 0.775i)3-s + (0.142 + 0.989i)5-s + (−1.80 + 3.94i)7-s + (0.277 − 1.92i)9-s + (−6.12 − 1.79i)11-s + (−0.327 − 0.716i)13-s + (0.671 − 0.775i)15-s + (−5.62 − 3.61i)17-s + (0.285 − 0.183i)19-s + (4.26 − 1.25i)21-s + (2.98 + 3.75i)23-s + (−0.959 + 0.281i)25-s + (−4.27 + 2.74i)27-s + (−3.76 − 2.41i)29-s + (−5.28 + 6.10i)31-s + ⋯
L(s)  = 1  + (−0.387 − 0.447i)3-s + (0.0636 + 0.442i)5-s + (−0.680 + 1.49i)7-s + (0.0923 − 0.642i)9-s + (−1.84 − 0.542i)11-s + (−0.0907 − 0.198i)13-s + (0.173 − 0.200i)15-s + (−1.36 − 0.876i)17-s + (0.0655 − 0.0421i)19-s + (0.931 − 0.273i)21-s + (0.622 + 0.782i)23-s + (−0.191 + 0.0563i)25-s + (−0.821 + 0.528i)27-s + (−0.698 − 0.449i)29-s + (−0.949 + 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.978 - 0.206i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.978 - 0.206i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0101089 + 0.0968188i\)
\(L(\frac12)\) \(\approx\) \(0.0101089 + 0.0968188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-2.98 - 3.75i)T \)
good3 \( 1 + (0.671 + 0.775i)T + (-0.426 + 2.96i)T^{2} \)
7 \( 1 + (1.80 - 3.94i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (6.12 + 1.79i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (0.327 + 0.716i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (5.62 + 3.61i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.285 + 0.183i)T + (7.89 - 17.2i)T^{2} \)
29 \( 1 + (3.76 + 2.41i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (5.28 - 6.10i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 + (0.127 - 0.886i)T + (-35.5 - 10.4i)T^{2} \)
41 \( 1 + (-0.293 - 2.04i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-2.23 - 2.58i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + (3.77 - 8.27i)T + (-34.7 - 40.0i)T^{2} \)
59 \( 1 + (5.63 + 12.3i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (-4.70 + 5.43i)T + (-8.68 - 60.3i)T^{2} \)
67 \( 1 + (-1.54 + 0.453i)T + (56.3 - 36.2i)T^{2} \)
71 \( 1 + (5.04 - 1.48i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-1.83 + 1.17i)T + (30.3 - 66.4i)T^{2} \)
79 \( 1 + (-0.756 - 1.65i)T + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (0.323 - 2.24i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-3.41 - 3.93i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + (-1.45 - 10.1i)T + (-93.0 + 27.3i)T^{2} \)
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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

−11.41825747918814343640410129387, −10.80226606373494396886884827171, −9.516440763962085018820116520392, −8.949578570417263443818744365235, −7.71552409430583491406105223694, −6.76215734676148812191963481409, −5.83550095615147535269911333997, −5.17389173777379361805819566683, −3.21504614640072639248056107552, −2.39687590777480535246968203987, 0.05732086137058969371431932891, 2.27703947632006885296477426591, 3.99987664108022744667616731916, 4.71977773662925490053387186446, 5.75268204837835552843286589563, 7.10683204901418482863314935786, 7.71309439152229082961570155580, 8.930305119329731122478840160975, 10.08600596195460838944339660489, 10.58198902065073601648214563812

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