Properties

Label 2-672-28.27-c1-0-7
Degree $2$
Conductor $672$
Sign $0.645 - 0.763i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + 3-s + 1.72i·5-s + (2.63 − 0.222i)7-s + 9-s + 6.17i·11-s − 2.82i·13-s + 1.72i·15-s + 1.72i·17-s − 5.90·19-s + (2.63 − 0.222i)21-s − 1.54i·23-s + 2.01·25-s + 27-s + 8.28·29-s − 4.62·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.772i·5-s + (0.996 − 0.0839i)7-s + 0.333·9-s + 1.86i·11-s − 0.784i·13-s + 0.446i·15-s + 0.419i·17-s − 1.35·19-s + (0.575 − 0.0484i)21-s − 0.322i·23-s + 0.402·25-s + 0.192·27-s + 1.53·29-s − 0.831·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.645 - 0.763i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.645 - 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78747 + 0.830046i\)
\(L(\frac12)\) \(\approx\) \(1.78747 + 0.830046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 + (-2.63 + 0.222i)T \)
good5 \( 1 - 1.72iT - 5T^{2} \)
11 \( 1 - 6.17iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 - 1.72iT - 17T^{2} \)
19 \( 1 + 5.90T + 19T^{2} \)
23 \( 1 + 1.54iT - 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 + 3.92iT - 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 - 4.88T + 47T^{2} \)
53 \( 1 + 9.17T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 14.2iT - 61T^{2} \)
67 \( 1 - 6.64iT - 67T^{2} \)
71 \( 1 + 5.00iT - 71T^{2} \)
73 \( 1 - 2.19iT - 73T^{2} \)
79 \( 1 + 9.55iT - 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + 9.16iT - 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−10.48639902129966050455285515679, −9.971460606208847162003545321272, −8.769796565661272033539943329129, −7.971397217151675363249551590115, −7.23274271973602203545682488095, −6.37967152947484092040596002930, −4.90306474515578374695848188095, −4.18785379853488530109142632889, −2.75015643463081279282559512106, −1.79950512818282886491283119469, 1.10515141901177421920431499057, 2.50090587673871485689066299179, 3.87711647648630139783688324353, 4.79427270100469632042395873401, 5.78807747692425263044720543962, 6.91157329428232996672360992649, 8.170017348563640631442494056281, 8.598678471663449436225664965461, 9.152437517870262720892920255849, 10.49720200608559343085033718750

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