Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.419 - 0.907i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.69i·3-s + (−23.4 − 50.7i)5-s + 10.2i·7-s + 220.·9-s − 596.·11-s − 420. i·13-s + (238. − 109. i)15-s + 974. i·17-s − 380.·19-s − 48.1·21-s − 3.54e3i·23-s + (−2.02e3 + 2.37e3i)25-s + 2.17e3i·27-s + 5.44e3·29-s − 3.62e3·31-s + ⋯
L(s)  = 1  + 0.301i·3-s + (−0.419 − 0.907i)5-s + 0.0791i·7-s + 0.909·9-s − 1.48·11-s − 0.690i·13-s + (0.273 − 0.126i)15-s + 0.817i·17-s − 0.241·19-s − 0.0238·21-s − 1.39i·23-s + (−0.648 + 0.760i)25-s + 0.574i·27-s + 1.20·29-s − 0.677·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.419 - 0.907i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.419 - 0.907i)\)
\(L(3)\)  \(\approx\)  \(0.6733017396\)
\(L(\frac12)\)  \(\approx\)  \(0.6733017396\)
\(L(\frac{7}{2})\)   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 \( 1 + (23.4 + 50.7i)T \)
good3 \( 1 - 4.69iT - 243T^{2} \)
7 \( 1 - 10.2iT - 1.68e4T^{2} \)
11 \( 1 + 596.T + 1.61e5T^{2} \)
13 \( 1 + 420. iT - 3.71e5T^{2} \)
17 \( 1 - 974. iT - 1.41e6T^{2} \)
19 \( 1 + 380.T + 2.47e6T^{2} \)
23 \( 1 + 3.54e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.44e3T + 2.05e7T^{2} \)
31 \( 1 + 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.75e3iT - 6.93e7T^{2} \)
41 \( 1 - 263.T + 1.15e8T^{2} \)
43 \( 1 - 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.34e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.34e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.90e3T + 7.14e8T^{2} \)
61 \( 1 + 2.94e4T + 8.44e8T^{2} \)
67 \( 1 + 7.16e3iT - 1.35e9T^{2} \)
71 \( 1 + 8.13e4T + 1.80e9T^{2} \)
73 \( 1 + 5.51e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 + 9.93e4T + 5.58e9T^{2} \)
97 \( 1 - 6.29e4iT - 8.58e9T^{2} \)
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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

−10.76653869529859718537887301479, −10.36847371849671746156380159455, −9.180391542920595754994519583565, −8.194790992081249211327176441642, −7.55695893516215936519822766378, −6.06459868227362595575809709827, −4.92764387936026587973165125001, −4.22053049540058034671384162238, −2.73574439866033537836486306785, −1.14687646472500973045576136169, 0.18948582447348967961414681821, 1.92706646031058121637840411138, 3.09287190966597809055576644506, 4.33623411511461873003633969814, 5.56460539700628240225327044176, 7.00958516277033845899059903322, 7.32090204813189579533585293135, 8.434419199142068710860258973928, 9.832337153291435846007941104182, 10.44890719774417605822813818525

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