Properties

Label 2-672-168.125-c1-0-18
Degree $2$
Conductor $672$
Sign $0.810 + 0.586i$
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  + (1.58 − 0.707i)3-s − 1.41i·5-s + (−1 + 2.44i)7-s + (2.00 − 2.23i)9-s + 3.46·11-s + 3.16·13-s + (−1.00 − 2.23i)15-s − 3.16·19-s + (0.150 + 4.58i)21-s − 4.47i·23-s + 2.99·25-s + (1.58 − 4.94i)27-s + 6.92·29-s − 4.89i·31-s + (5.47 − 2.44i)33-s + ⋯
L(s)  = 1  + (0.912 − 0.408i)3-s − 0.632i·5-s + (−0.377 + 0.925i)7-s + (0.666 − 0.745i)9-s + 1.04·11-s + 0.877·13-s + (−0.258 − 0.577i)15-s − 0.725·19-s + (0.0329 + 0.999i)21-s − 0.932i·23-s + 0.599·25-s + (0.304 − 0.952i)27-s + 1.28·29-s − 0.879i·31-s + (0.953 − 0.426i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.810 + 0.586i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.810 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03041 - 0.657254i\)
\(L(\frac12)\) \(\approx\) \(2.03041 - 0.657254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.707i)T \)
7 \( 1 + (1 - 2.44i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 3.16T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 - 7.74iT - 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 + 7.74iT - 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 - 14.6iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 7.07iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−10.16697798389915916581490516402, −9.286756655768727599167840285219, −8.597241764143249324910319585389, −8.241938050060657769911635753194, −6.67664083794175110510025654016, −6.26569680475358989837615124618, −4.77555314275788396650016768619, −3.71336544659971089523523125935, −2.57869259744024151210033242552, −1.28255966979167215658691597815, 1.57542542556234187223751856720, 3.22206064920503753287867575028, 3.74142265716364723103502605634, 4.84253023928328590788741073590, 6.52286787416596006104724700285, 6.93125501136967075748077644411, 8.129504764314362475123073568164, 8.848898928084947787865995532529, 9.796955015460360831141853796321, 10.48055475546877640196931799702

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