Properties

Degree 2
Conductor $ 5 \cdot 11 $
Sign $0.983 + 0.180i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.15 − 1.15i)3-s + 2i·4-s + (−1.65 − 1.5i)5-s + 0.316i·9-s − 3.31·11-s + (2.31 + 2.31i)12-s + (−3.65 + 0.183i)15-s − 4·16-s + (3 − 3.31i)20-s + (6.15 − 6.15i)23-s + (0.5 + 4.97i)25-s + (3.84 + 3.84i)27-s + 9.94·31-s + (−3.84 + 3.84i)33-s − 0.633·36-s + (−8.47 − 8.47i)37-s + ⋯
L(s)  = 1  + (0.668 − 0.668i)3-s + i·4-s + (−0.741 − 0.670i)5-s + 0.105i·9-s − 1.00·11-s + (0.668 + 0.668i)12-s + (−0.944 + 0.0473i)15-s − 16-s + (0.670 − 0.741i)20-s + (1.28 − 1.28i)23-s + (0.100 + 0.994i)25-s + (0.739 + 0.739i)27-s + 1.78·31-s + (−0.668 + 0.668i)33-s − 0.105·36-s + (−1.39 − 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $0.983 + 0.180i$
motivic weight  =  \(1\)
character  :  $\chi_{55} (32, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 55,\ (\ :1/2),\ 0.983 + 0.180i)$
$L(1)$  $\approx$  $0.904114 - 0.0823837i$
$L(\frac12)$  $\approx$  $0.904114 - 0.0823837i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11\}$, \(F_p(T)\) is a polynomial of degree 2. If $p \in \{5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad5 \( 1 + (1.65 + 1.5i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 - 2iT^{2} \)
3 \( 1 + (-1.15 + 1.15i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-6.15 + 6.15i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + (8.47 + 8.47i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (2.68 + 2.68i)T + 47iT^{2} \)
53 \( 1 + (9.63 - 9.63i)T - 53iT^{2} \)
59 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (13.4 + 13.4i)T + 97iT^{2} \)
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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

−15.46383208499094676616911500459, −13.89001808452755247293335082366, −12.90018437961591414739990847332, −12.32524711808401709605989414965, −10.88138989192355342832196302463, −8.813174501455526830475471380197, −8.083045099302244928948101749412, −7.17128803580651590665461554075, −4.69283492377536908539427929709, −2.85991361577896401400185087078, 3.14810190884510550788584453401, 4.90586862298137220451443600131, 6.70998683208428034530949216352, 8.278663877808189586967777173745, 9.682303072155755329859385347862, 10.53863727231625196018524193354, 11.66991069215909497491782236321, 13.47315814693768272499698520822, 14.54757051505721561843252976477, 15.41924105522343092015745758834

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