Properties

Degree $2$
Conductor $672$
Sign $-0.169 + 0.985i$
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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.28447 - 1.52361i\)
\(L(\frac12)\) \(\approx\) \(1.28447 - 1.52361i\)
\(L(\frac{3}{2})\) 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

−9.796012052781630190187509904504, −9.195275975483909467784840210835, −8.568021392915641977384319891711, −7.995810697959164256697538490677, −6.74910554981350092535816366830, −5.77761464814791610263279151894, −4.58153993771419100457007259389, −3.77133077181238926323244474587, −2.06505240811029814122507348554, −1.05578848175629554390963341314, 2.15295781254606517118379312516, 3.23365520787058157105723438170, 3.83731975708371451139929277015, 5.19273371515305651271300810021, 6.73012105073072881703946465272, 7.07984132504543770430997395876, 8.038223960623643298242847950683, 9.306160842258568765817756687645, 9.857581637363525441335210536503, 10.62425752592734083047865656216

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