Properties

Degree $2$
Conductor $1734$
Sign $0.638 - 0.769i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (0.707 + 0.707i)3-s − 4-s + (1.41 + 1.41i)5-s + (0.707 − 0.707i)6-s + i·8-s + 1.00i·9-s + (1.41 − 1.41i)10-s + (−2.82 + 2.82i)11-s + (−0.707 − 0.707i)12-s + 2·13-s + 2.00i·15-s + 16-s + 1.00·18-s − 4i·19-s + (−1.41 − 1.41i)20-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.632 + 0.632i)5-s + (0.288 − 0.288i)6-s + 0.353i·8-s + 0.333i·9-s + (0.447 − 0.447i)10-s + (−0.852 + 0.852i)11-s + (−0.204 − 0.204i)12-s + 0.554·13-s + 0.516i·15-s + 0.250·16-s + 0.235·18-s − 0.917i·19-s + (−0.316 − 0.316i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1734\)    =    \(2 \cdot 3 \cdot 17^{2}\)
Sign: $0.638 - 0.769i$
Motivic weight: \(1\)
Character: $\chi_{1734} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1734,\ (\ :1/2),\ 0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875569103\)
\(L(\frac12)\) \(\approx\) \(1.875569103\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 \)
good5 \( 1 + (-1.41 - 1.41i)T + 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + (-7.07 - 7.07i)T + 29iT^{2} \)
31 \( 1 + (-5.65 - 5.65i)T + 31iT^{2} \)
37 \( 1 + (1.41 + 1.41i)T + 37iT^{2} \)
41 \( 1 + (7.07 - 7.07i)T - 41iT^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + (-7.07 + 7.07i)T - 61iT^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (7.07 + 7.07i)T + 73iT^{2} \)
79 \( 1 + (5.65 - 5.65i)T - 79iT^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-9.89 - 9.89i)T + 97iT^{2} \)
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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

−9.627569212591744685361818828340, −8.779513316601854980470917439544, −8.088453963684844770298565225182, −7.01744557893090597620161380983, −6.24323155993572064555889166174, −5.00037074411665176561237703695, −4.52141574697682164516372558356, −3.07830398946111315135958691862, −2.71123168396102281928516274806, −1.47636988186842666570485247348, 0.69169905610264995419003434904, 2.02802793126179874867724678641, 3.25782635837158200272772839711, 4.31154389756146238155898685732, 5.46395910157928627437454571796, 5.88669135276707711730224983634, 6.78065634721167287798466715701, 7.78727571047300743346359690419, 8.435123725518845428766956404192, 8.825010810829502556297122199962

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