Properties

Degree $2$
Conductor $672$
Sign $-0.386 + 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 13-s + (1 − 1.73i)17-s + (−2.5 − 4.33i)19-s + (−2 + 1.73i)21-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 0.999·27-s − 8·29-s + (1.5 − 2.59i)31-s + (−0.999 − 1.73i)33-s + (4.5 + 7.79i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 0.277·13-s + (0.242 − 0.420i)17-s + (−0.573 − 0.993i)19-s + (−0.436 + 0.377i)21-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 0.192·27-s − 1.48·29-s + (0.269 − 0.466i)31-s + (−0.174 − 0.301i)33-s + (0.739 + 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.386 + 0.922i$
Motivic weight: \(1\)
Character: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665899 - 1.00107i\)
\(L(\frac12)\) \(\approx\) \(0.665899 - 1.00107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−10.18987665732156992179273476391, −9.289119926915806724691452792594, −8.547433564021256449521646652138, −7.57842880539696012350789495096, −6.58152171388148417110033479455, −6.08374990537256173628167383456, −4.57356004989035895150065611371, −3.45402249761594588939019684739, −2.41844532142829202541071407104, −0.60812738537787409802359400123, 1.89900277113488870219400588468, 3.35911810172884238495230737964, 4.03603809203216524183816860156, 5.44262333822350199102950112321, 6.20206266565479074778989239588, 7.32630130891329094292412796946, 8.245822510331323119501921809955, 9.355376221145531575895543091884, 9.660380412515403207056694063490, 10.67769103457568103418758291920

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