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Properties

Label 2-672-168.101-c1-0-17
Degree $2$
Conductor $672$
Sign $0.870 - 0.491i$
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  + (1.5 + 0.866i)3-s + (−0.621 + 0.358i)5-s + (2.62 + 0.358i)7-s + (1.5 + 2.59i)9-s + (2.91 − 5.04i)11-s − 1.24·15-s + (3.62 + 2.80i)21-s + (−2.24 + 3.88i)25-s + 5.19i·27-s − 7.58·29-s + (9.62 + 5.55i)31-s + (8.74 − 5.04i)33-s + (−1.75 + 0.717i)35-s + (−1.86 − 1.07i)45-s + (6.74 + 1.88i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.277 + 0.160i)5-s + (0.990 + 0.135i)7-s + (0.5 + 0.866i)9-s + (0.878 − 1.52i)11-s − 0.320·15-s + (0.790 + 0.612i)21-s + (−0.448 + 0.776i)25-s + 0.999i·27-s − 1.40·29-s + (1.72 + 0.997i)31-s + (1.52 − 0.878i)33-s + (−0.297 + 0.121i)35-s + (−0.277 − 0.160i)45-s + (0.963 + 0.268i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.870 - 0.491i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.870 - 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13124 + 0.560512i\)
\(L(\frac12)\) \(\approx\) \(2.13124 + 0.560512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 + (-2.62 - 0.358i)T \)
good5 \( 1 + (0.621 - 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.91 + 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + (-9.62 - 5.55i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.03 + 3.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.9 + 7.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.86 + 15.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.5iT - 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.06iT - 97T^{2} \)
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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

−10.69007475041549822905849875446, −9.540203843719486853021143231231, −8.747151441455426521058949486021, −8.182330239286997046352751271829, −7.31379558293676093406956442827, −6.02922910124575637344521453271, −4.93670492334240298154299997817, −3.89317499024493505577282416227, −3.05061270706615394995318809565, −1.55416361376436812678291719203, 1.40667237725378205320687868986, 2.41005208670120327193575356706, 4.01888894305330788164185027004, 4.57401618642052231341101273731, 6.12696773276472184504705824930, 7.22980091920701204937054134655, 7.74265486223848597575391910580, 8.625764852447604011237988708515, 9.469588114412123537313569081449, 10.23259890897280968010745826422

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