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Properties

Label 2-460-115.107-c1-0-10
Degree $2$
Conductor $460$
Sign $-0.904 + 0.427i$
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  + (−2.30 − 0.859i)3-s + (−0.944 − 2.02i)5-s + (4.12 − 0.897i)7-s + (2.30 + 1.99i)9-s + (−1.49 + 0.215i)11-s + (0.152 − 0.699i)13-s + (0.434 + 5.48i)15-s + (2.28 − 4.19i)17-s + (−7.47 − 2.19i)19-s + (−10.2 − 1.47i)21-s + (−3.93 + 2.73i)23-s + (−3.21 + 3.82i)25-s + (−0.0566 − 0.103i)27-s + (−2.29 − 7.82i)29-s + (−0.903 + 1.97i)31-s + ⋯
L(s)  = 1  + (−1.33 − 0.496i)3-s + (−0.422 − 0.906i)5-s + (1.55 − 0.339i)7-s + (0.767 + 0.665i)9-s + (−0.451 + 0.0649i)11-s + (0.0422 − 0.194i)13-s + (0.112 + 1.41i)15-s + (0.555 − 1.01i)17-s + (−1.71 − 0.503i)19-s + (−2.24 − 0.322i)21-s + (−0.820 + 0.571i)23-s + (−0.643 + 0.765i)25-s + (−0.0108 − 0.0199i)27-s + (−0.426 − 1.45i)29-s + (−0.162 + 0.355i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.904 + 0.427i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135432 - 0.603478i\)
\(L(\frac12)\) \(\approx\) \(0.135432 - 0.603478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.944 + 2.02i)T \)
23 \( 1 + (3.93 - 2.73i)T \)
good3 \( 1 + (2.30 + 0.859i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (-4.12 + 0.897i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (1.49 - 0.215i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.152 + 0.699i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (-2.28 + 4.19i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (7.47 + 2.19i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (2.29 + 7.82i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.903 - 1.97i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-0.203 - 2.84i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (7.15 + 8.25i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (0.844 - 2.26i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (-0.303 - 0.303i)T + 47iT^{2} \)
53 \( 1 + (0.657 + 3.02i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (0.467 + 0.727i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (-12.1 - 5.55i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (2.50 + 3.34i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (0.850 - 5.91i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-8.23 + 4.49i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (13.1 - 8.44i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-9.22 + 0.659i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (2.15 + 4.71i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (3.03 + 0.217i)T + (96.0 + 13.8i)T^{2} \)
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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

−11.03948463330895658549704620236, −10.02231633724385466613007311609, −8.588576724436628415527128984824, −7.893568726462677493234264249257, −7.02701339619313295479021125406, −5.68837629242251315274844278058, −5.03563101474173140690393454716, −4.21914485869461840966294709755, −1.82622021867895142046629826943, −0.45140937772940947833989412036, 1.99149541192372288633245937160, 3.88538260471825735132740585088, 4.80871213672949770478430791124, 5.76814916061918371904574601401, 6.57692284733668889689720705324, 7.87827298473324549994025292537, 8.514600228550172143808029395855, 10.25272529586656111110256895833, 10.64886336973572078633799629835, 11.28898114378308669374528224616

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