Properties

Degree 2
Conductor $ 2^{5} \cdot 5 $
Sign $-0.327 + 0.944i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 + 5.65i)3-s + (−54.5 + 12.2i)5-s + (23.8 + 23.8i)7-s + 178. i·9-s + 218. i·11-s + (152. + 152. i)13-s + (239. − 377. i)15-s + (318. − 318. i)17-s − 2.45e3·19-s − 270.·21-s + (−512. + 512. i)23-s + (2.82e3 − 1.33e3i)25-s + (−2.38e3 − 2.38e3i)27-s − 5.94e3i·29-s − 3.06e3i·31-s + ⋯
L(s)  = 1  + (−0.362 + 0.362i)3-s + (−0.975 + 0.218i)5-s + (0.184 + 0.184i)7-s + 0.736i·9-s + 0.543i·11-s + (0.250 + 0.250i)13-s + (0.274 − 0.433i)15-s + (0.267 − 0.267i)17-s − 1.56·19-s − 0.133·21-s + (−0.201 + 0.201i)23-s + (0.904 − 0.426i)25-s + (−0.630 − 0.630i)27-s − 1.31i·29-s − 0.572i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(160\)    =    \(2^{5} \cdot 5\)
\( \varepsilon \)  =  $-0.327 + 0.944i$
motivic weight  =  \(5\)
character  :  $\chi_{160} (63, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 160,\ (\ :5/2),\ -0.327 + 0.944i)\)
\(L(3)\)  \(\approx\)  \(0.2031982984\)
\(L(\frac12)\)  \(\approx\)  \(0.2031982984\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (54.5 - 12.2i)T \)
good3 \( 1 + (5.65 - 5.65i)T - 243iT^{2} \)
7 \( 1 + (-23.8 - 23.8i)T + 1.68e4iT^{2} \)
11 \( 1 - 218. iT - 1.61e5T^{2} \)
13 \( 1 + (-152. - 152. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-318. + 318. i)T - 1.41e6iT^{2} \)
19 \( 1 + 2.45e3T + 2.47e6T^{2} \)
23 \( 1 + (512. - 512. i)T - 6.43e6iT^{2} \)
29 \( 1 + 5.94e3iT - 2.05e7T^{2} \)
31 \( 1 + 3.06e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.31e3 + 1.31e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 6.09e3T + 1.15e8T^{2} \)
43 \( 1 + (1.90e3 - 1.90e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (8.04e3 + 8.04e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (8.09e3 + 8.09e3i)T + 4.18e8iT^{2} \)
59 \( 1 + 4.16e4T + 7.14e8T^{2} \)
61 \( 1 - 4.30e4T + 8.44e8T^{2} \)
67 \( 1 + (4.16e4 + 4.16e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 2.37e4iT - 1.80e9T^{2} \)
73 \( 1 + (-9.33e3 - 9.33e3i)T + 2.07e9iT^{2} \)
79 \( 1 - 8.60e4T + 3.07e9T^{2} \)
83 \( 1 + (7.58e4 - 7.58e4i)T - 3.93e9iT^{2} \)
89 \( 1 + 1.86e4iT - 5.58e9T^{2} \)
97 \( 1 + (9.77e4 - 9.77e4i)T - 8.58e9iT^{2} \)
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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

−11.52588917452838317190659810524, −10.85122196745045760150584306232, −9.810557974241622678315088807493, −8.405533603099569304854176718766, −7.59467141091238567406657100949, −6.29314190778181634666137027475, −4.84984879584204614728079019114, −3.96306213843512112418577752820, −2.21195161939312623989040451545, −0.07853856617321036398884190249, 1.17902911520192284746270255018, 3.30197122661423958202731523617, 4.45272523790869236097599612995, 5.95887373528243185852748298491, 6.99514316700601065257057660552, 8.162649870892493268103780005965, 8.985791723321332173726275077215, 10.58309178915557732961039435681, 11.31593902773784012962868887646, 12.38083367578473348642282786318

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