Properties

Degree 2
Conductor $ 2^{2} \cdot 5 $
Sign $0.309 - 0.951i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.61 + 1.17i)2-s + 3.80i·3-s + (1.23 − 3.80i)4-s + 2.23·5-s + (−4.47 − 6.15i)6-s − 8.50i·7-s + (2.47 + 7.60i)8-s − 5.47·9-s + (−3.61 + 2.62i)10-s + 1.79i·11-s + (14.4 + 4.70i)12-s + 0.472·13-s + (10 + 13.7i)14-s + 8.50i·15-s + (−12.9 − 9.40i)16-s − 23.8·17-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + 1.26i·3-s + (0.309 − 0.951i)4-s + 0.447·5-s + (−0.745 − 1.02i)6-s − 1.21i·7-s + (0.309 + 0.951i)8-s − 0.608·9-s + (−0.361 + 0.262i)10-s + 0.163i·11-s + (1.20 + 0.391i)12-s + 0.0363·13-s + (0.714 + 0.983i)14-s + 0.567i·15-s + (−0.809 − 0.587i)16-s − 1.40·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $0.309 - 0.951i$
motivic weight  =  \(2\)
character  :  $\chi_{20} (11, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 20,\ (\ :1),\ 0.309 - 0.951i)$
$L(\frac{3}{2})$  $\approx$  $0.529541 + 0.384734i$
$L(\frac12)$  $\approx$  $0.529541 + 0.384734i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + (1.61 - 1.17i)T \)
5 \( 1 - 2.23T \)
good3 \( 1 - 3.80iT - 9T^{2} \)
7 \( 1 + 8.50iT - 49T^{2} \)
11 \( 1 - 1.79iT - 121T^{2} \)
13 \( 1 - 0.472T + 169T^{2} \)
17 \( 1 + 23.8T + 289T^{2} \)
19 \( 1 + 9.40iT - 361T^{2} \)
23 \( 1 + 16.1iT - 529T^{2} \)
29 \( 1 - 6.94T + 841T^{2} \)
31 \( 1 - 47.4iT - 961T^{2} \)
37 \( 1 - 26.3T + 1.36e3T^{2} \)
41 \( 1 + 41.4T + 1.68e3T^{2} \)
43 \( 1 + 2.00iT - 1.84e3T^{2} \)
47 \( 1 + 35.3iT - 2.20e3T^{2} \)
53 \( 1 + 21.6T + 2.80e3T^{2} \)
59 \( 1 - 73.8iT - 3.48e3T^{2} \)
61 \( 1 + 26.1T + 3.72e3T^{2} \)
67 \( 1 - 88.8iT - 4.48e3T^{2} \)
71 \( 1 + 39.4iT - 5.04e3T^{2} \)
73 \( 1 - 137.T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 - 21.2iT - 6.88e3T^{2} \)
89 \( 1 - 67.4T + 7.92e3T^{2} \)
97 \( 1 + 39.1T + 9.40e3T^{2} \)
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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

−18.00161076901848455689995019023, −16.97977800438529860839386420923, −16.06473506443160315819183234820, −14.95973834450253858699656403971, −13.66048308923934595117485624503, −10.88861831861723776689182975920, −10.13985424320236302314452955734, −8.865872522679871094283943224714, −6.85329683828389638656960119829, −4.68474825363585902296547097666, 2.16603142707187025273094328595, 6.39638741873142387487651468247, 8.062354773420145265675760261931, 9.399943457714330104094633546675, 11.35638204847778194796798066602, 12.48887786565000405871751767505, 13.46485836927477789336601076842, 15.54352874446815682561493753686, 17.23214691949979215025745013403, 18.24990748626544848423266845060

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