Properties

Degree $2$
Conductor $1470$
Sign $0.0332 + 0.999i$
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 i·3-s − 4-s + (0.0743 + 2.23i)5-s + 6-s i·8-s − 9-s + (−2.23 + 0.0743i)10-s − 3.05·11-s + i·12-s − 1.64i·13-s + (2.23 − 0.0743i)15-s + 16-s − 2.90i·17-s i·18-s − 2.21·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.0332 + 0.999i)5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + (−0.706 + 0.0234i)10-s − 0.921·11-s + 0.288i·12-s − 0.455i·13-s + (0.577 − 0.0191i)15-s + 0.250·16-s − 0.705i·17-s − 0.235i·18-s − 0.507·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0332 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1470\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.0332 + 0.999i$
Motivic weight: \(1\)
Character: $\chi_{1470} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1470,\ (\ :1/2),\ 0.0332 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5692516406\)
\(L(\frac12)\) \(\approx\) \(0.5692516406\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-0.0743 - 2.23i)T \)
7 \( 1 \)
good11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 1.64iT - 13T^{2} \)
17 \( 1 + 2.90iT - 17T^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + 1.78iT - 23T^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 - 0.944T + 31T^{2} \)
37 \( 1 + 10.7iT - 37T^{2} \)
41 \( 1 - 6.67T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 + 5.78iT - 53T^{2} \)
59 \( 1 - 5.97T + 59T^{2} \)
61 \( 1 - 0.445T + 61T^{2} \)
67 \( 1 - 1.26iT - 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 + 7.91iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 0.874iT - 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.169276386761043777431871860951, −8.280061606476861413166631783311, −7.36191938292366892914269017856, −7.19638250525956792280814080284, −5.99806069947475825725080912104, −5.54935973515901176817526583334, −4.28313088894110301366934751338, −3.09763064380491924083325948108, −2.19405761644086089522953361220, −0.22392332452715152657293592922, 1.43568227392360777057977869079, 2.61889768955266914783535128808, 3.82500222971617153509863546480, 4.54745974282710837851294522486, 5.31899434026911482743885601043, 6.14533347447115095349604330374, 7.58253693592236392169693824627, 8.403818355999432805485668859813, 8.954092017592008916664725590849, 9.849006961894563041536467396170

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