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Properties

Label 2-460-115.102-c1-0-8
Degree $2$
Conductor $460$
Sign $0.932 + 0.360i$
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  + (2.90 − 0.632i)3-s + (2.22 − 0.165i)5-s + (−2.67 + 1.46i)7-s + (5.32 − 2.43i)9-s + (−3.76 − 3.26i)11-s + (2.01 − 3.69i)13-s + (6.37 − 1.89i)15-s + (4.84 + 3.62i)17-s + (−0.805 + 5.60i)19-s + (−6.85 + 5.93i)21-s + (−4.62 + 1.25i)23-s + (4.94 − 0.739i)25-s + (6.78 − 5.08i)27-s + (−5.05 + 0.727i)29-s + (1.91 − 1.23i)31-s + ⋯
L(s)  = 1  + (1.67 − 0.365i)3-s + (0.997 − 0.0741i)5-s + (−1.01 + 0.552i)7-s + (1.77 − 0.810i)9-s + (−1.13 − 0.983i)11-s + (0.560 − 1.02i)13-s + (1.64 − 0.488i)15-s + (1.17 + 0.879i)17-s + (−0.184 + 1.28i)19-s + (−1.49 + 1.29i)21-s + (−0.965 + 0.260i)23-s + (0.988 − 0.147i)25-s + (1.30 − 0.977i)27-s + (−0.939 + 0.135i)29-s + (0.343 − 0.220i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.932 + 0.360i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.932 + 0.360i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43573 - 0.454072i\)
\(L(\frac12)\) \(\approx\) \(2.43573 - 0.454072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.165i)T \)
23 \( 1 + (4.62 - 1.25i)T \)
good3 \( 1 + (-2.90 + 0.632i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (2.67 - 1.46i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (3.76 + 3.26i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-2.01 + 3.69i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-4.84 - 3.62i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.805 - 5.60i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (5.05 - 0.727i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-1.91 + 1.23i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (7.63 + 2.84i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (1.13 - 2.48i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-2.60 - 11.9i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-0.0341 - 0.0341i)T + 47iT^{2} \)
53 \( 1 + (0.981 + 1.79i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (1.02 + 3.49i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-0.389 - 0.605i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (1.18 + 0.0849i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (2.29 + 2.64i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (6.24 + 8.34i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-3.04 + 0.893i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-5.51 + 14.7i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-4.76 - 3.06i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (4.21 + 11.3i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.52071300025479878232804638924, −9.981706060304068283654999551773, −9.152415781095939191144556386228, −8.201929052859098121739081342189, −7.81304697108816093375368093491, −6.13886112988261715848722410937, −5.67820945243340421593552696343, −3.48491555068480471784526295770, −2.99984034671347806971055852920, −1.73332269270058070334760424986, 2.04797403375832905762615412684, 2.93023427070471897710371929331, 4.01985853343446594612283270811, 5.24991170238197694338054325994, 6.80696422007384060656317733807, 7.41481433473439321855134688732, 8.644688027629124201661748007025, 9.435202242473158960725028483250, 9.937111663605146086002429368510, 10.57891940518168611462896911454

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