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Properties

Label 2-672-168.101-c1-0-19
Degree $2$
Conductor $672$
Sign $-0.0326 + 0.999i$
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  + (−0.609 − 1.62i)3-s + (−2.24 + 1.29i)5-s + (2.53 + 0.751i)7-s + (−2.25 + 1.97i)9-s + (1.63 − 2.83i)11-s + 0.912·13-s + (3.47 + 2.85i)15-s + (2.39 − 4.14i)17-s + (−2.66 − 4.61i)19-s + (−0.326 − 4.57i)21-s + (4.45 − 2.57i)23-s + (0.870 − 1.50i)25-s + (4.57 + 2.45i)27-s − 1.35·29-s + (−8.18 − 4.72i)31-s + ⋯
L(s)  = 1  + (−0.351 − 0.936i)3-s + (−1.00 + 0.580i)5-s + (0.958 + 0.284i)7-s + (−0.752 + 0.658i)9-s + (0.493 − 0.855i)11-s + 0.253·13-s + (0.897 + 0.737i)15-s + (0.580 − 1.00i)17-s + (−0.611 − 1.05i)19-s + (−0.0712 − 0.997i)21-s + (0.929 − 0.536i)23-s + (0.174 − 0.301i)25-s + (0.881 + 0.473i)27-s − 0.252·29-s + (−1.47 − 0.848i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0326 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0326 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.0326 + 0.999i$
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: \((2,\ 672,\ (\ :1/2),\ -0.0326 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752555 - 0.777518i\)
\(L(\frac12)\) \(\approx\) \(0.752555 - 0.777518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.609 + 1.62i)T \)
7 \( 1 + (-2.53 - 0.751i)T \)
good5 \( 1 + (2.24 - 1.29i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.912T + 13T^{2} \)
17 \( 1 + (-2.39 + 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.66 + 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.45 + 2.57i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + (8.18 + 4.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.59 + 0.922i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.91T + 41T^{2} \)
43 \( 1 + 8.00iT - 43T^{2} \)
47 \( 1 + (3.29 + 5.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.841 - 1.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.50 - 0.867i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.72 + 8.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 6.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.603iT - 71T^{2} \)
73 \( 1 + (1.29 + 0.746i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0625 - 0.108i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.246iT - 83T^{2} \)
89 \( 1 + (-1.80 - 3.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.10iT - 97T^{2} \)
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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

−10.87231523012788540055264927647, −9.122053310162273666610007242327, −8.403960961462822766900353667460, −7.50645677319093223844617394439, −6.98192528150683557731122475957, −5.85162595295861353834283950652, −4.88872585067824496081686506041, −3.54729724064595925945607354559, −2.33180771338702198120720207730, −0.67247282458021800038314761325, 1.43325343767091402917635357613, 3.59548262223836058396588850223, 4.24534994441780410609708001424, 5.00875722414094223602281126240, 6.06942835052424110746160710329, 7.42914408875188279843336344280, 8.197118479536771655443836797463, 8.968818472516545879145029157482, 9.930761208824099546211289597556, 10.87033157246335916576584092060

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