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Properties

Label 2-672-21.17-c1-0-24
Degree $2$
Conductor $672$
Sign $-0.895 + 0.444i$
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  − 1.73·3-s + (−0.5 − 0.866i)5-s + (1.73 − 2i)7-s + 2.99·9-s + (−4.33 − 2.5i)11-s + 3.46i·13-s + (0.866 + 1.49i)15-s + (−0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−2.99 + 3.46i)21-s + (−4.33 + 2.5i)23-s + (2 − 3.46i)25-s − 5.19·27-s − 3.46i·29-s + (−7.79 − 4.5i)31-s + ⋯
L(s)  = 1  − 1.00·3-s + (−0.223 − 0.387i)5-s + (0.654 − 0.755i)7-s + 0.999·9-s + (−1.30 − 0.753i)11-s + 0.960i·13-s + (0.223 + 0.387i)15-s + (−0.121 + 0.210i)17-s + (0.198 − 0.114i)19-s + (−0.654 + 0.755i)21-s + (−0.902 + 0.521i)23-s + (0.400 − 0.692i)25-s − 1.00·27-s − 0.643i·29-s + (−1.39 − 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101550 - 0.433491i\)
\(L(\frac12)\) \(\approx\) \(0.101550 - 0.433491i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73T \)
7 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.33 + 2.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 - 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.46iT - 29T^{2} \)
31 \( 1 + (7.79 + 4.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.5 + 6.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.866 - 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−10.40602900554137594703296855616, −9.415061680508884957187870578904, −8.171767128272051233188596746841, −7.56567377567572924432667711568, −6.50465166870952610464025306998, −5.49302041441799309490506587364, −4.70267326941240199478200948030, −3.78835519489972937040244494181, −1.84395372839307911294820081728, −0.26123722547201962325188344290, 1.85001151088652100575298502003, 3.24760287114128486590638123730, 4.91471594233920015700779617363, 5.22888760136331035188951355902, 6.34062073996813680506728900141, 7.43726242189230491381760796311, 8.011960259321632685666459425490, 9.265460809364602051655111886692, 10.39210540796154114160246236509, 10.74592394244891863408669442662

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