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Properties

Label 2-672-168.11-c1-0-20
Degree $2$
Conductor $672$
Sign $-0.643 + 0.765i$
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  + (−1.03 − 1.38i)3-s + (0.692 + 1.19i)5-s + (−2.08 − 1.62i)7-s + (−0.839 + 2.88i)9-s + (3.82 + 2.20i)11-s − 6.43i·13-s + (0.941 − 2.20i)15-s + (−2.52 − 1.45i)17-s + (−1.58 − 2.75i)19-s + (−0.0791 + 4.58i)21-s + (−1.84 − 3.19i)23-s + (1.54 − 2.67i)25-s + (4.86 − 1.83i)27-s − 6.67·29-s + (−2.17 − 1.25i)31-s + ⋯
L(s)  = 1  + (−0.600 − 0.799i)3-s + (0.309 + 0.536i)5-s + (−0.789 − 0.613i)7-s + (−0.279 + 0.960i)9-s + (1.15 + 0.665i)11-s − 1.78i·13-s + (0.243 − 0.569i)15-s + (−0.613 − 0.354i)17-s + (−0.364 − 0.631i)19-s + (−0.0172 + 0.999i)21-s + (−0.385 − 0.667i)23-s + (0.308 − 0.534i)25-s + (0.935 − 0.352i)27-s − 1.24·29-s + (−0.390 − 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.643 + 0.765i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351184 - 0.753733i\)
\(L(\frac12)\) \(\approx\) \(0.351184 - 0.753733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 + 1.38i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
good5 \( 1 + (-0.692 - 1.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.82 - 2.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.43iT - 13T^{2} \)
17 \( 1 + (2.52 + 1.45i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.75i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.84 + 3.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.67T + 29T^{2} \)
31 \( 1 + (2.17 + 1.25i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.00 - 2.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.497iT - 41T^{2} \)
43 \( 1 + 0.865T + 43T^{2} \)
47 \( 1 + (1.59 + 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.12 + 7.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.62 + 3.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.99 + 1.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.36 + 5.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.90T + 71T^{2} \)
73 \( 1 + (-3.23 + 5.60i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.65 - 0.953i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.00iT - 83T^{2} \)
89 \( 1 + (-8.22 + 4.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.19T + 97T^{2} \)
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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

−10.37919625458392665018397664150, −9.497377422769538782592505412947, −8.308512068744784932022145823448, −7.23312716789803712359411964160, −6.69995268777595029401315812455, −5.96608186527134342021164980550, −4.77453809440819716364627298282, −3.40839349751616857051582128904, −2.15511793472806675148263606668, −0.47129250826797268990888328788, 1.71380078856180904625846779373, 3.57256343303906118053060970656, 4.24964531797280380953600697767, 5.53035517243133058092845582102, 6.18336010833217222349824879600, 6.96870488055644391216359261477, 8.867707548956388370814204939924, 9.037450483786402488502203180611, 9.743356410904863145460411800791, 10.89634866941042231619797093010

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