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Properties

Label 2-453-453.50-c0-0-0
Degree $2$
Conductor $453$
Sign $0.973 + 0.230i$
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  + (0.728 − 0.684i)3-s + (0.309 + 0.951i)4-s + (−0.456 − 0.718i)7-s + (0.0627 − 0.998i)9-s + (0.876 + 0.481i)12-s + (−0.0534 − 0.113i)13-s + (−0.809 + 0.587i)16-s + (0.598 + 1.84i)19-s + (−0.824 − 0.211i)21-s + (−0.992 − 0.125i)25-s + (−0.637 − 0.770i)27-s + (0.542 − 0.656i)28-s + (−0.996 + 0.394i)31-s + (0.968 − 0.248i)36-s + (−0.620 − 1.31i)37-s + ⋯
L(s)  = 1  + (0.728 − 0.684i)3-s + (0.309 + 0.951i)4-s + (−0.456 − 0.718i)7-s + (0.0627 − 0.998i)9-s + (0.876 + 0.481i)12-s + (−0.0534 − 0.113i)13-s + (−0.809 + 0.587i)16-s + (0.598 + 1.84i)19-s + (−0.824 − 0.211i)21-s + (−0.992 − 0.125i)25-s + (−0.637 − 0.770i)27-s + (0.542 − 0.656i)28-s + (−0.996 + 0.394i)31-s + (0.968 − 0.248i)36-s + (−0.620 − 1.31i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(453\)    =    \(3 \cdot 151\)
Sign: $0.973 + 0.230i$
Analytic conductor: \(0.226076\)
Root analytic conductor: \(0.475474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{453} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 453,\ (\ :0),\ 0.973 + 0.230i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066278483\)
\(L(\frac12)\) \(\approx\) \(1.066278483\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.728 + 0.684i)T \)
151 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.992 + 0.125i)T^{2} \)
7 \( 1 + (0.456 + 0.718i)T + (-0.425 + 0.904i)T^{2} \)
11 \( 1 + (-0.0627 - 0.998i)T^{2} \)
13 \( 1 + (0.0534 + 0.113i)T + (-0.637 + 0.770i)T^{2} \)
17 \( 1 + (0.425 + 0.904i)T^{2} \)
19 \( 1 + (-0.598 - 1.84i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.929 - 0.368i)T^{2} \)
31 \( 1 + (0.996 - 0.394i)T + (0.728 - 0.684i)T^{2} \)
37 \( 1 + (0.620 + 1.31i)T + (-0.637 + 0.770i)T^{2} \)
41 \( 1 + (-0.968 + 0.248i)T^{2} \)
43 \( 1 + (0.200 - 0.316i)T + (-0.425 - 0.904i)T^{2} \)
47 \( 1 + (-0.968 + 0.248i)T^{2} \)
53 \( 1 + (-0.535 + 0.844i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.17 + 1.10i)T + (0.0627 + 0.998i)T^{2} \)
67 \( 1 + (-0.110 - 1.74i)T + (-0.992 + 0.125i)T^{2} \)
71 \( 1 + (0.425 - 0.904i)T^{2} \)
73 \( 1 + (-0.331 - 0.521i)T + (-0.425 + 0.904i)T^{2} \)
79 \( 1 + (-1.84 - 0.730i)T + (0.728 + 0.684i)T^{2} \)
83 \( 1 + (-0.876 - 0.481i)T^{2} \)
89 \( 1 + (-0.876 - 0.481i)T^{2} \)
97 \( 1 + (0.101 + 1.61i)T + (-0.992 + 0.125i)T^{2} \)
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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

−11.47878223549252250960134459909, −10.28558023621108807430934838062, −9.355820746478680650244983550902, −8.300186357801880117083129615732, −7.61047176316991741749802719700, −6.97306241726755056638597553891, −5.86611889024054448614352351783, −3.91804109694894800022395346417, −3.35831716457718182325110235821, −1.91681145185990929842250150677, 2.11402466544849394990064177644, 3.18792114001954230880093534067, 4.69371561750947074136715658477, 5.51847677602389967890194713002, 6.63755741078755329947012676709, 7.72640629099365215180881110821, 9.151200829855189725843786181703, 9.300586644394961918126260858905, 10.34423103336024855168342324453, 11.14401461571433489091220440315

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