Properties

Degree $2$
Conductor $672$
Sign $-0.985 - 0.169i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + i)3-s − 4.24i·5-s i·7-s + (1.00 − 2.82i)9-s − 4.24·11-s − 2·13-s + (4.24 + 6i)15-s + 7.07i·17-s + 4i·19-s + (1 + 1.41i)21-s − 1.41·23-s − 12.9·25-s + (1.41 + 5.00i)27-s − 2.82i·29-s + 2i·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s − 1.89i·5-s − 0.377i·7-s + (0.333 − 0.942i)9-s − 1.27·11-s − 0.554·13-s + (1.09 + 1.54i)15-s + 1.71i·17-s + 0.917i·19-s + (0.218 + 0.308i)21-s − 0.294·23-s − 2.59·25-s + (0.272 + 0.962i)27-s − 0.525i·29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.985 - 0.169i$
Motivic weight: \(1\)
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0147008 + 0.172617i\)
\(L(\frac12)\) \(\approx\) \(0.0147008 + 0.172617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - i)T \)
7 \( 1 + iT \)
good5 \( 1 + 4.24iT - 5T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.21243468138869474116625072403, −9.246909995444855125658604275900, −8.369102173266163936169729450352, −7.63270450414219960239135295993, −6.05341147610731636537069524960, −5.40127792819419315531938345137, −4.60688793857510330588028211194, −3.80927786191459989162801470643, −1.62486225674077431030278790305, −0.098004205654109376909010208787, 2.38709511202510936307703961529, 2.94612821573295632240837478873, 4.81431042286693696984655232716, 5.68120859663521373439814683102, 6.71117244513687385833961969281, 7.25479816294770423799084573660, 7.923983490879210221172410033010, 9.543186762232093089563253591387, 10.32037215320400078705683431431, 11.06570780712792484820267987193

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