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Properties

Label 2-672-168.101-c1-0-11
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

\(L(1)\) \(\approx\) \(1.16561 + 0.306554i\)
\(L(\frac12)\) \(\approx\) \(1.16561 + 0.306554i\)
\(L(\frac{3}{2})\) 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

−10.41502202505027086925638067519, −10.12110709230228102390127041665, −8.719805012416717061047778401213, −7.76685475944534842093583275622, −7.13256520472491480462266831997, −6.04744429158877863800721790269, −4.98481475203830793227457046205, −4.59610847647984613228342281586, −2.46617499730207489343222664795, −1.38505331364018386573061687479, 0.808786369348333386255620335300, 2.67877329282256188867752941831, 4.08274524128991227999016179734, 5.04525844219978013053040846727, 5.82976806317769829452527945115, 6.64211017116379267075769928475, 8.011687591575841840991814963112, 8.556982559816063838656324293728, 9.941634497219394253254593457044, 10.42727663750207501905600228972

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