Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.883 + 0.468i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24.8·3-s + (−53.4 − 16.3i)5-s − 100. i·7-s + 372.·9-s − 5.36i·11-s − 506.·13-s + (−1.32e3 − 406. i)15-s − 1.45e3i·17-s + 722. i·19-s − 2.48e3i·21-s + 3.08e3i·23-s + (2.58e3 + 1.75e3i)25-s + 3.20e3·27-s − 7.29e3i·29-s − 8.60e3·31-s + ⋯
L(s)  = 1  + 1.59·3-s + (−0.956 − 0.293i)5-s − 0.773i·7-s + 1.53·9-s − 0.0133i·11-s − 0.831·13-s + (−1.52 − 0.466i)15-s − 1.22i·17-s + 0.459i·19-s − 1.23i·21-s + 1.21i·23-s + (0.828 + 0.560i)25-s + 0.846·27-s − 1.61i·29-s − 1.60·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.883 + 0.468i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.883 + 0.468i)\)
\(L(3)\)  \(\approx\)  \(1.352144581\)
\(L(\frac12)\)  \(\approx\)  \(1.352144581\)
\(L(\frac{7}{2})\)   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 \( 1 + (53.4 + 16.3i)T \)
good3 \( 1 - 24.8T + 243T^{2} \)
7 \( 1 + 100. iT - 1.68e4T^{2} \)
11 \( 1 + 5.36iT - 1.61e5T^{2} \)
13 \( 1 + 506.T + 3.71e5T^{2} \)
17 \( 1 + 1.45e3iT - 1.41e6T^{2} \)
19 \( 1 - 722. iT - 2.47e6T^{2} \)
23 \( 1 - 3.08e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.29e3iT - 2.05e7T^{2} \)
31 \( 1 + 8.60e3T + 2.86e7T^{2} \)
37 \( 1 + 5.27e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 + 8.34e3T + 1.47e8T^{2} \)
47 \( 1 + 2.21e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.33e4T + 4.18e8T^{2} \)
59 \( 1 - 1.09e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.09e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.30e4T + 1.35e9T^{2} \)
71 \( 1 - 3.60e4T + 1.80e9T^{2} \)
73 \( 1 + 6.42e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.86e4T + 3.07e9T^{2} \)
83 \( 1 + 4.97e4T + 3.93e9T^{2} \)
89 \( 1 - 7.76e3T + 5.58e9T^{2} \)
97 \( 1 - 3.68e4iT - 8.58e9T^{2} \)
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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

−10.10170749390243935522057374490, −9.368610933860315211338448681592, −8.432452017541406878608645521286, −7.53875010547911849522648949770, −7.16906083555839657214078596318, −5.07412183372687089276033574018, −3.91871329216541684689077206338, −3.23855615431135178203301253527, −1.86374653166404933977668404552, −0.26080891014079971409091142885, 1.86325758900006500940121914949, 2.92320367427630662931586514785, 3.74204636379407217170905185573, 4.97011244680936673503771751045, 6.70068108773810657900065193894, 7.59411710147936331520477105836, 8.517914899669658559262882371991, 8.931892098140783734543736944562, 10.13602329441708379858529932508, 11.11653928455750449962273028069

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