Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 \cdot 11 $
Sign $-0.870 - 0.492i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (1 + 1.41i)3-s i·5-s + (−1.41 − 2.00i)6-s i·7-s + 2.82·8-s + (−1.00 + 2.82i)9-s + 1.41i·10-s + (−3 + 1.41i)11-s − 6.24i·13-s + 1.41i·14-s + (1.41 − i)15-s − 4.00·16-s − 0.171·17-s + (1.41 − 4.00i)18-s + 5.24i·19-s + ⋯
L(s)  = 1  − 1.00·2-s + (0.577 + 0.816i)3-s − 0.447i·5-s + (−0.577 − 0.816i)6-s − 0.377i·7-s + 0.999·8-s + (−0.333 + 0.942i)9-s + 0.447i·10-s + (−0.904 + 0.426i)11-s − 1.73i·13-s + 0.377i·14-s + (0.365 − 0.258i)15-s − 1.00·16-s − 0.0416·17-s + (0.333 − 0.942i)18-s + 1.20i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-0.870 - 0.492i$
motivic weight  =  \(1\)
character  :  $\chi_{1155} (1121, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1155,\ (\ :1/2),\ -0.870 - 0.492i)\)
\(L(1)\)  \(\approx\)  \(0.4527423979\)
\(L(\frac12)\)  \(\approx\)  \(0.4527423979\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;11\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (3 - 1.41i)T \)
good2 \( 1 + 1.41T + 2T^{2} \)
13 \( 1 + 6.24iT - 13T^{2} \)
17 \( 1 + 0.171T + 17T^{2} \)
19 \( 1 - 5.24iT - 19T^{2} \)
23 \( 1 - 7.24iT - 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
31 \( 1 + 0.242T + 31T^{2} \)
37 \( 1 + 0.242T + 37T^{2} \)
41 \( 1 + 7.41T + 41T^{2} \)
43 \( 1 - 5.24iT - 43T^{2} \)
47 \( 1 - 13.0iT - 47T^{2} \)
53 \( 1 + 1.58iT - 53T^{2} \)
59 \( 1 - 6.89iT - 59T^{2} \)
61 \( 1 - 0.757iT - 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 2.24iT - 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.995259322398124904836472461543, −9.481153703904381174760094350748, −8.442544382379299109342295347445, −7.915650443076854751973526706339, −7.48531947704859778106705494876, −5.65339590942926893933508536536, −5.00448469238100960787864606433, −3.99769320202935130891428941274, −2.95791552281750598186005063717, −1.45682568409534002257902270674, 0.26149358816505158561078031671, 1.86924436097575112259862729840, 2.67971866236220210564290020060, 4.06983312236944761954448656367, 5.22339212245830237446049623334, 6.73041019965430853882589890832, 6.91203704590542942370675355705, 8.078927569222376065701099474376, 8.640896599696818699266394564433, 9.183000035098062507217929303383

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