Properties

Degree $2$
Conductor $643$
Sign $-1$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s − 1.41i·19-s + 1.41i·21-s − 2.00·22-s + ⋯
L(s)  = 1  + 1.41i·2-s + 1.41i·3-s − 1.00·4-s − 2.00·6-s + 7-s − 1.00·9-s + 1.41i·11-s − 1.41i·12-s − 1.41i·13-s + 1.41i·14-s − 0.999·16-s − 1.41i·17-s − 1.41i·18-s − 1.41i·19-s + 1.41i·21-s − 2.00·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(643\)
Sign: $-1$
Motivic weight: \(0\)
Character: $\chi_{643} (642, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 643,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9998375976\)
\(L(\frac12)\) \(\approx\) \(0.9998375976\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 + T \)
good2 \( 1 - 1.41iT - T^{2} \)
3 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.94851248684766075150090021217, −10.21073057568331177393524905809, −9.295957548365579215927861060039, −8.588436502787797536151879370326, −7.57229765495950296068733716981, −6.97469559691433318812029989732, −5.46230021353657668148743583308, −4.96123848042881506906606405373, −4.36523896901306282917837482643, −2.66937142137420542033697244843, 1.42930776884257491251972020313, 1.90117538784150014303107356247, 3.35015415167390916554184317334, 4.42456215706889537249931396120, 6.00730813306307802748592261812, 6.69237568284240319215992247555, 8.089968740191711208081140194445, 8.398714907456713505586374525086, 9.640466990688581648898876419575, 10.83393620990277159714687536110

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