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Properties

Label 2-455-455.363-c1-0-34
Degree $2$
Conductor $455$
Sign $0.501 + 0.865i$
Analytic cond. $3.63319$
Root an. cond. $1.90609$
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 + 0.707i)2-s + (−0.618 + 0.618i)3-s − 0.999i·4-s + (−2.12 + 0.707i)5-s − 0.874·6-s + (−1.58 − 2.12i)7-s + (2.12 − 2.12i)8-s + 2.23i·9-s + (−2 − 0.999i)10-s − 5.45i·11-s + (0.618 + 0.618i)12-s + (3.58 − 0.418i)13-s + (0.381 − 2.61i)14-s + (0.874 − 1.74i)15-s + 1.00·16-s + (−0.763 − 0.763i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.356 + 0.356i)3-s − 0.499i·4-s + (−0.948 + 0.316i)5-s − 0.356·6-s + (−0.597 − 0.801i)7-s + (0.750 − 0.750i)8-s + 0.745i·9-s + (−0.632 − 0.316i)10-s − 1.64i·11-s + (0.178 + 0.178i)12-s + (0.993 − 0.116i)13-s + (0.102 − 0.699i)14-s + (0.225 − 0.451i)15-s + 0.250·16-s + (−0.185 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(455\)    =    \(5 \cdot 7 \cdot 13\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(3.63319\)
Root analytic conductor: \(1.90609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{455} (363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 455,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.954578 - 0.550150i\)
\(L(\frac12)\) \(\approx\) \(0.954578 - 0.550150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.12 - 0.707i)T \)
7 \( 1 + (1.58 + 2.12i)T \)
13 \( 1 + (-3.58 + 0.418i)T \)
good2 \( 1 + (-0.707 - 0.707i)T + 2iT^{2} \)
3 \( 1 + (0.618 - 0.618i)T - 3iT^{2} \)
11 \( 1 + 5.45iT - 11T^{2} \)
17 \( 1 + (0.763 + 0.763i)T + 17iT^{2} \)
19 \( 1 + 6.86iT - 19T^{2} \)
23 \( 1 + (-0.381 - 0.381i)T + 23iT^{2} \)
29 \( 1 - 0.763iT - 29T^{2} \)
31 \( 1 + 5.78T + 31T^{2} \)
37 \( 1 + (-3.16 - 3.16i)T + 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (0.854 + 0.854i)T + 43iT^{2} \)
47 \( 1 + (2.82 - 2.82i)T - 47iT^{2} \)
53 \( 1 + (-8.23 - 8.23i)T + 53iT^{2} \)
59 \( 1 + 1.20iT - 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 + (3.70 + 3.70i)T + 67iT^{2} \)
71 \( 1 - 8.27iT - 71T^{2} \)
73 \( 1 + (-10.0 - 10.0i)T + 73iT^{2} \)
79 \( 1 + 9.70iT - 79T^{2} \)
83 \( 1 + (-9.69 - 9.69i)T + 83iT^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 + (-7.40 + 7.40i)T - 97iT^{2} \)
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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

−11.03348300890963322558902507398, −10.38444176476421780591817310272, −9.100647103044053547820200370361, −8.004713257757903307151293805767, −6.99670260022680194951757744936, −6.23257163160463661854137032389, −5.19087899083075284902171854489, −4.18462685313326156031214635035, −3.24071735688840595248175712218, −0.62673920435039368153581413614, 1.86514144278583823812981050217, 3.47600239514948681677293101984, 4.10484915979576390227675399126, 5.40450388007677167513836146770, 6.60836867360474048394681376908, 7.53092273632793157885044152823, 8.476224335638739165536183848611, 9.374786435169089849075587696028, 10.58370332165349252023516018473, 11.80548837757998952878129058871

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