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Properties

Label 2-453-453.425-c0-0-0
Degree $2$
Conductor $453$
Sign $-0.988 - 0.152i$
Analytic cond. $0.226076$
Root an. cond. $0.475474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.637 − 0.770i)3-s + (−0.809 − 0.587i)4-s + (−1.92 + 0.242i)7-s + (−0.187 + 0.982i)9-s + (0.0627 + 0.998i)12-s + (−0.362 − 0.0931i)13-s + (0.309 + 0.951i)16-s + (−1.17 − 0.856i)19-s + (1.41 + 1.32i)21-s + (−0.929 − 0.368i)25-s + (0.876 − 0.481i)27-s + (1.69 + 0.933i)28-s + (0.844 − 1.79i)31-s + (0.728 − 0.684i)36-s + (−1.23 − 0.317i)37-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)3-s + (−0.809 − 0.587i)4-s + (−1.92 + 0.242i)7-s + (−0.187 + 0.982i)9-s + (0.0627 + 0.998i)12-s + (−0.362 − 0.0931i)13-s + (0.309 + 0.951i)16-s + (−1.17 − 0.856i)19-s + (1.41 + 1.32i)21-s + (−0.929 − 0.368i)25-s + (0.876 − 0.481i)27-s + (1.69 + 0.933i)28-s + (0.844 − 1.79i)31-s + (0.728 − 0.684i)36-s + (−1.23 − 0.317i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ -0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1641434721\)
\(L(\frac12)\) \(\approx\) \(0.1641434721\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.637 + 0.770i)T \)
151 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.929 + 0.368i)T^{2} \)
7 \( 1 + (1.92 - 0.242i)T + (0.968 - 0.248i)T^{2} \)
11 \( 1 + (0.187 + 0.982i)T^{2} \)
13 \( 1 + (0.362 + 0.0931i)T + (0.876 + 0.481i)T^{2} \)
17 \( 1 + (-0.968 - 0.248i)T^{2} \)
19 \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.425 - 0.904i)T^{2} \)
31 \( 1 + (-0.844 + 1.79i)T + (-0.637 - 0.770i)T^{2} \)
37 \( 1 + (1.23 + 0.317i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (-0.728 + 0.684i)T^{2} \)
43 \( 1 + (1.06 + 0.134i)T + (0.968 + 0.248i)T^{2} \)
47 \( 1 + (-0.728 + 0.684i)T^{2} \)
53 \( 1 + (0.992 + 0.125i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.393 - 0.476i)T + (-0.187 - 0.982i)T^{2} \)
67 \( 1 + (0.0235 + 0.123i)T + (-0.929 + 0.368i)T^{2} \)
71 \( 1 + (-0.968 + 0.248i)T^{2} \)
73 \( 1 + (-1.60 + 0.202i)T + (0.968 - 0.248i)T^{2} \)
79 \( 1 + (-0.791 - 1.68i)T + (-0.637 + 0.770i)T^{2} \)
83 \( 1 + (-0.0627 - 0.998i)T^{2} \)
89 \( 1 + (-0.0627 - 0.998i)T^{2} \)
97 \( 1 + (0.115 + 0.607i)T + (-0.929 + 0.368i)T^{2} \)
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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

−10.68390942547329165042798489330, −9.932266780893901075358889003576, −9.197912039235026565583411760500, −8.118981985462546552566955492849, −6.75159944586928206032286888549, −6.23468245151728526513147728852, −5.31585629034719053469398155495, −4.01378591626668796436903640894, −2.41166692791553024309080987588, −0.22217067236388223334165070435, 3.22633322187131746119901120693, 3.86236462306786536868536268714, 4.99306407604054799147488528047, 6.17450792976784485257484589699, 6.94650727671697340484367733704, 8.399108175957838792164458664528, 9.296748919471651896200972030055, 9.969352453518573556933802210756, 10.54412450643276243903456794155, 12.06824957203262768614090368695

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