Properties

Degree 2
Conductor $ 2^{6} \cdot 5 $
Sign $-0.0894 + 0.995i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.1i·3-s + (5 − 55.6i)5-s − 122. i·7-s + 119.·9-s − 100·11-s + 734. i·13-s + (619. + 55.6i)15-s − 979. i·17-s + 2.24e3·19-s + 1.36e3·21-s − 3.41e3i·23-s + (−3.07e3 − 556. i)25-s + 4.03e3i·27-s − 7.85e3·29-s + 2.14e3·31-s + ⋯
L(s)  = 1  + 0.714i·3-s + (0.0894 − 0.995i)5-s − 0.944i·7-s + 0.489·9-s − 0.249·11-s + 1.20i·13-s + (0.711 + 0.0638i)15-s − 0.822i·17-s + 1.42·19-s + 0.674·21-s − 1.34i·23-s + (−0.983 − 0.178i)25-s + 1.06i·27-s − 1.73·29-s + 0.400·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.0894 + 0.995i$
motivic weight  =  \(5\)
character  :  $\chi_{320} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 320,\ (\ :5/2),\ -0.0894 + 0.995i)\)
\(L(3)\)  \(\approx\)  \(1.650248077\)
\(L(\frac12)\)  \(\approx\)  \(1.650248077\)
\(L(\frac{7}{2})\)   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 \( 1 + (-5 + 55.6i)T \)
good3 \( 1 - 11.1iT - 243T^{2} \)
7 \( 1 + 122. iT - 1.68e4T^{2} \)
11 \( 1 + 100T + 1.61e5T^{2} \)
13 \( 1 - 734. iT - 3.71e5T^{2} \)
17 \( 1 + 979. iT - 1.41e6T^{2} \)
19 \( 1 - 2.24e3T + 2.47e6T^{2} \)
23 \( 1 + 3.41e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.85e3T + 2.05e7T^{2} \)
31 \( 1 - 2.14e3T + 2.86e7T^{2} \)
37 \( 1 - 1.04e4iT - 6.93e7T^{2} \)
41 \( 1 + 7.41e3T + 1.15e8T^{2} \)
43 \( 1 + 1.77e4iT - 1.47e8T^{2} \)
47 \( 1 + 9.43e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.42e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.59e4T + 7.14e8T^{2} \)
61 \( 1 - 3.05e3T + 8.44e8T^{2} \)
67 \( 1 + 5.87e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.76e4T + 1.80e9T^{2} \)
73 \( 1 + 2.40e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.97e4T + 3.07e9T^{2} \)
83 \( 1 + 1.62e4iT - 3.93e9T^{2} \)
89 \( 1 - 826T + 5.58e9T^{2} \)
97 \( 1 - 3.75e4iT - 8.58e9T^{2} \)
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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

−10.36073297503395336830007795927, −9.621212577586265929609035630620, −8.936388524099825809113371961115, −7.66247122572529922476196792655, −6.78003575548190241237761128212, −5.19399462906579573142919958004, −4.52887319864426694364326652379, −3.59094956574471590014844025913, −1.70459240082522058207036972744, −0.44340848046082226308189162315, 1.37045245245271877766990722539, 2.56114965143684325321073482418, 3.59536784670161724108992805416, 5.46181142286028004731511694339, 6.10064657929294140039396587696, 7.45625845686601697938979634600, 7.77157959699295499897780676075, 9.271758284613131533267663081524, 10.12581781821141615021545329504, 11.11574585690471616448469564112

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