Properties

Degree 2
Conductor $ 5 \cdot 11 $
Sign $0.278 + 0.960i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 − 2.15i)3-s − 2i·4-s + (1.65 + 1.5i)5-s + 6.31i·9-s + 3.31·11-s + (−4.31 + 4.31i)12-s + (−0.341 − 6.81i)15-s − 4·16-s + (3 − 3.31i)20-s + (2.84 + 2.84i)23-s + (0.5 + 4.97i)25-s + (7.15 − 7.15i)27-s − 9.94·31-s + (−7.15 − 7.15i)33-s + 12.6·36-s + (1.47 − 1.47i)37-s + ⋯
L(s)  = 1  + (−1.24 − 1.24i)3-s i·4-s + (0.741 + 0.670i)5-s + 2.10i·9-s + 1.00·11-s + (−1.24 + 1.24i)12-s + (−0.0882 − 1.76i)15-s − 16-s + (0.670 − 0.741i)20-s + (0.592 + 0.592i)23-s + (0.100 + 0.994i)25-s + (1.37 − 1.37i)27-s − 1.78·31-s + (−1.24 − 1.24i)33-s + 2.10·36-s + (0.242 − 0.242i)37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(55\)    =    \(5 \cdot 11\)
\( \varepsilon \)  =  $0.278 + 0.960i$
motivic weight  =  \(1\)
character  :  $\chi_{55} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 55,\ (\ :1/2),\ 0.278 + 0.960i)$
$L(1)$  $\approx$  $0.547687 - 0.411573i$
$L(\frac12)$  $\approx$  $0.547687 - 0.411573i$
$L(\frac{3}{2})$   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 \( 1 + (2.15 + 2.15i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-2.84 - 2.84i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.94T + 31T^{2} \)
37 \( 1 + (-1.47 + 1.47i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (9.31 - 9.31i)T - 47iT^{2} \)
53 \( 1 + (-3.63 - 3.63i)T + 53iT^{2} \)
59 \( 1 - 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-11.4 + 11.4i)T - 67iT^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (3.52 - 3.52i)T - 97iT^{2} \)
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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

−14.84950796954489427052169666659, −13.89276280389374815506058098110, −12.88317184914451630183209063433, −11.48835924044194409813873677265, −10.80504971422507305312653999049, −9.444252028395126395372971013627, −7.15190906861610606750797165916, −6.31419279479992790599764988959, −5.38481502662120442712443357971, −1.63620250690544105536133033406, 3.94953060036178666445540372141, 5.18591295389126206683905173421, 6.60354026877879547142645090208, 8.796465424989386929398989851816, 9.704271736043385903489154722322, 11.06498465641040350573303438256, 12.02674481896044705874260636110, 13.00672764986797603071920136975, 14.63565379371261967531007266738, 16.06944287166255069982915769266

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