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Properties

Label 2-460-115.103-c1-0-6
Degree $2$
Conductor $460$
Sign $0.981 - 0.192i$
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  + (−2.32 + 1.74i)3-s + (1.79 + 1.33i)5-s + (0.283 − 3.96i)7-s + (1.53 − 5.24i)9-s + (2.54 − 3.95i)11-s + (1.13 − 0.0813i)13-s + (−6.50 + 0.00446i)15-s + (−1.63 + 4.39i)17-s + (0.754 − 1.65i)19-s + (6.25 + 9.73i)21-s + (4.76 + 0.497i)23-s + (1.41 + 4.79i)25-s + (2.50 + 6.71i)27-s + (6.44 − 2.94i)29-s + (0.271 + 1.88i)31-s + ⋯
L(s)  = 1  + (−1.34 + 1.00i)3-s + (0.800 + 0.598i)5-s + (0.107 − 1.50i)7-s + (0.513 − 1.74i)9-s + (0.766 − 1.19i)11-s + (0.315 − 0.0225i)13-s + (−1.67 + 0.00115i)15-s + (−0.397 + 1.06i)17-s + (0.173 − 0.379i)19-s + (1.36 + 2.12i)21-s + (0.994 + 0.103i)23-s + (0.283 + 0.959i)25-s + (0.481 + 1.29i)27-s + (1.19 − 0.546i)29-s + (0.0487 + 0.339i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08464 + 0.105650i\)
\(L(\frac12)\) \(\approx\) \(1.08464 + 0.105650i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 - 1.33i)T \)
23 \( 1 + (-4.76 - 0.497i)T \)
good3 \( 1 + (2.32 - 1.74i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.283 + 3.96i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (-2.54 + 3.95i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-1.13 + 0.0813i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (1.63 - 4.39i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-0.754 + 1.65i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-6.44 + 2.94i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.271 - 1.88i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (2.54 + 4.65i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (-2.37 + 0.698i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (3.45 + 4.61i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (8.76 - 8.76i)T - 47iT^{2} \)
53 \( 1 + (-8.15 - 0.583i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (1.06 + 0.926i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-5.40 + 0.777i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-14.7 - 3.20i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-8.58 + 5.51i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (4.86 - 1.81i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (3.92 - 4.52i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (9.26 - 5.05i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-1.39 + 9.73i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (6.71 + 3.66i)T + (52.4 + 81.6i)T^{2} \)
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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

−11.02730730259641925365415534488, −10.40690993422918182158062518346, −9.706256389114564769622165283132, −8.573889061191397237722649315846, −6.92313802116634051202257508623, −6.36540801787607427090782637805, −5.45532250293614211976324712263, −4.30207785841364572302209116555, −3.44912746493381267799761570247, −0.986763448416868822204803745131, 1.30495925872684819511725273018, 2.37129491371252233662683357356, 4.88329399635145319801374086183, 5.32402432946561154961006011596, 6.41431445120080515768452875175, 6.90683005755843857206647054155, 8.365760041595188873856613768224, 9.249843375238834428893316759526, 10.12555653813404805127560003834, 11.51207655236987239554301774909

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