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Properties

Label 2-464-29.24-c1-0-1
Degree $2$
Conductor $464$
Sign $-0.712 - 0.701i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
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  + (0.760 + 3.33i)3-s + (1.00 + 1.26i)5-s + (−0.338 − 1.48i)7-s + (−7.81 + 3.76i)9-s + (1.70 + 0.822i)11-s + (2.87 + 1.38i)13-s + (−3.44 + 4.31i)15-s − 1.69·17-s + (−0.818 + 3.58i)19-s + (4.68 − 2.25i)21-s + (1.67 − 2.10i)23-s + (0.531 − 2.33i)25-s + (−12.1 − 15.1i)27-s + (−4.31 − 3.21i)29-s + (1.11 + 1.39i)31-s + ⋯
L(s)  = 1  + (0.439 + 1.92i)3-s + (0.450 + 0.564i)5-s + (−0.127 − 0.560i)7-s + (−2.60 + 1.25i)9-s + (0.515 + 0.248i)11-s + (0.796 + 0.383i)13-s + (−0.888 + 1.11i)15-s − 0.412·17-s + (−0.187 + 0.822i)19-s + (1.02 − 0.492i)21-s + (0.349 − 0.438i)23-s + (0.106 − 0.466i)25-s + (−2.32 − 2.92i)27-s + (−0.801 − 0.597i)29-s + (0.199 + 0.250i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615477 + 1.50203i\)
\(L(\frac12)\) \(\approx\) \(0.615477 + 1.50203i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (4.31 + 3.21i)T \)
good3 \( 1 + (-0.760 - 3.33i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.00 - 1.26i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.338 + 1.48i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-1.70 - 0.822i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.87 - 1.38i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 1.69T + 17T^{2} \)
19 \( 1 + (0.818 - 3.58i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (-1.67 + 2.10i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-1.11 - 1.39i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-6.37 + 3.06i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + 5.03T + 41T^{2} \)
43 \( 1 + (-1.18 + 1.48i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.31 - 1.11i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-5.14 - 6.44i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 2.95T + 59T^{2} \)
61 \( 1 + (-1.72 - 7.55i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-2.23 + 1.07i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-5.46 - 2.63i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.33 + 2.93i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + (-8.30 + 4.00i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (1.22 - 5.38i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-9.95 - 12.4i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (3.31 - 14.5i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.85743464102314270297358537952, −10.53063768887285285672017130198, −9.641449398601057299927872607666, −9.007950727387200401026629290701, −8.049827910076534516127810892828, −6.56531993905283643740252528931, −5.59410865504235806617607765887, −4.28517537281962612865865388242, −3.77610315713645486682598900420, −2.48738137143071201227512970315, 1.01703864079630608611094691626, 2.14555504329320380835915004101, 3.33795268404707396111420087610, 5.36868356401212993050070282498, 6.20120128095594458551448710312, 6.96203569374746893776100307101, 8.011025390220776620630465657335, 8.834696460599497613628830459364, 9.279651858090733368149530468775, 11.11855370648164043237860577776

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