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Properties

Label 2-460-460.459-c1-0-35
Degree $2$
Conductor $460$
Sign $0.994 - 0.106i$
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  + (0.707 + 1.22i)2-s − 2.82·3-s + (−0.999 + 1.73i)4-s + 2.23·5-s + (−2.00 − 3.46i)6-s − 3.87i·7-s − 2.82·8-s + 5.00·9-s + (1.58 + 2.73i)10-s + (2.82 − 4.89i)12-s − 4.89i·13-s + (4.74 − 2.73i)14-s − 6.32·15-s + (−2.00 − 3.46i)16-s − 2.23·17-s + (3.53 + 6.12i)18-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s − 1.63·3-s + (−0.499 + 0.866i)4-s + 0.999·5-s + (−0.816 − 1.41i)6-s − 1.46i·7-s − 0.999·8-s + 1.66·9-s + (0.500 + 0.866i)10-s + (0.816 − 1.41i)12-s − 1.35i·13-s + (1.26 − 0.731i)14-s − 1.63·15-s + (−0.500 − 0.866i)16-s − 0.542·17-s + (0.833 + 1.44i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11274 + 0.0596848i\)
\(L(\frac12)\) \(\approx\) \(1.11274 + 0.0596848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
5 \( 1 - 2.23T \)
23 \( 1 + (-2.82 - 3.87i)T \)
good3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 + 3.87iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 + 2.23T + 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 7.74iT - 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 + 8.66iT - 59T^{2} \)
61 \( 1 + 10.9iT - 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + 1.73iT - 71T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + 3.87iT - 83T^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−11.06721103277099793188670137121, −10.27556634735442067454990931725, −9.524084534665901549612029911497, −7.901547387737075946089764760375, −7.02097876453095018274364778666, −6.33941122029207920052374973013, −5.34958859786892044150814436614, −4.88761383156983986650690269603, −3.44177427987399969342995058303, −0.811455823849379578805745272629, 1.44694200950501164191740648905, 2.70811422547149365974081392905, 4.67675692465143368471771516070, 5.22905382107307555511504664470, 6.13232594606970816605369461203, 6.64287164649789717146742872779, 8.866660886792576055208631036786, 9.518248902593101507744187303194, 10.35682218903678647027859665895, 11.38366816362505578629550287263

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