Properties

Degree $2$
Conductor $160$
Sign $0.389 + 0.920i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.47 + 1.47i)3-s + (55.2 + 8.52i)5-s + (156. − 156. i)7-s − 238. i·9-s − 35.4i·11-s + (−247. + 247. i)13-s + (69.0 + 94.2i)15-s + (−1.19e3 − 1.19e3i)17-s − 1.01e3·19-s + 461.·21-s + (2.03e3 + 2.03e3i)23-s + (2.97e3 + 942. i)25-s + (711. − 711. i)27-s − 2.20e3i·29-s − 6.17e3i·31-s + ⋯
L(s)  = 1  + (0.0947 + 0.0947i)3-s + (0.988 + 0.152i)5-s + (1.20 − 1.20i)7-s − 0.982i·9-s − 0.0884i·11-s + (−0.405 + 0.405i)13-s + (0.0792 + 0.108i)15-s + (−1.00 − 1.00i)17-s − 0.642·19-s + 0.228·21-s + (0.801 + 0.801i)23-s + (0.953 + 0.301i)25-s + (0.187 − 0.187i)27-s − 0.487i·29-s − 1.15i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.389 + 0.920i$
Motivic weight: \(5\)
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.389 + 0.920i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.492505446\)
\(L(\frac12)\) \(\approx\) \(2.492505446\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-55.2 - 8.52i)T \)
good3 \( 1 + (-1.47 - 1.47i)T + 243iT^{2} \)
7 \( 1 + (-156. + 156. i)T - 1.68e4iT^{2} \)
11 \( 1 + 35.4iT - 1.61e5T^{2} \)
13 \( 1 + (247. - 247. i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.19e3 + 1.19e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 1.01e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.03e3 - 2.03e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 2.20e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.17e3iT - 2.86e7T^{2} \)
37 \( 1 + (9.46e3 + 9.46e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 9.00e3T + 1.15e8T^{2} \)
43 \( 1 + (-1.59e4 - 1.59e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-7.19e3 + 7.19e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.29e4 - 1.29e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 4.05e4T + 7.14e8T^{2} \)
61 \( 1 - 2.92e4T + 8.44e8T^{2} \)
67 \( 1 + (-1.82e4 + 1.82e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.66e4iT - 1.80e9T^{2} \)
73 \( 1 + (-3.64e4 + 3.64e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 5.08e3T + 3.07e9T^{2} \)
83 \( 1 + (-2.03e4 - 2.03e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 5.93e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.95e4 - 6.95e4i)T + 8.58e9iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.56543955315306713484195001746, −10.84488493134352286905921037450, −9.719867152481433492651542434124, −8.901732547901923429064955022673, −7.44376874399030134724102309059, −6.56961915285117737989583067064, −5.11147925617091287673683831735, −3.99562826290536771647064027847, −2.21796798113967994700151704038, −0.819059916348767244595269756700, 1.71961416883680084948431854474, 2.46320163188241689481934650008, 4.80672884243087302335856843164, 5.44667599984356612209801126057, 6.80146407523161995586810302047, 8.438486334148420902779161275909, 8.730621887931952302780800602294, 10.32982970653936797272857069406, 11.00323219282556043062918914572, 12.33558258175009710750465817305

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