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Properties

Label 2-320-16.11-c2-0-1
Degree $2$
Conductor $320$
Sign $-0.177 - 0.984i$
Analytic cond. $8.71936$
Root an. cond. $2.95285$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 − 1.91i)3-s + (−1.58 − 1.58i)5-s − 1.82·7-s − 1.67i·9-s + (−8.81 + 8.81i)11-s + (−6.87 + 6.87i)13-s + 6.05i·15-s + 18.7·17-s + (3.65 + 3.65i)19-s + (3.48 + 3.48i)21-s + 3.89·23-s + 5.00i·25-s + (−20.4 + 20.4i)27-s + (−28.8 + 28.8i)29-s + 58.4i·31-s + ⋯
L(s)  = 1  + (−0.637 − 0.637i)3-s + (−0.316 − 0.316i)5-s − 0.260·7-s − 0.186i·9-s + (−0.801 + 0.801i)11-s + (−0.528 + 0.528i)13-s + 0.403i·15-s + 1.10·17-s + (0.192 + 0.192i)19-s + (0.165 + 0.165i)21-s + 0.169·23-s + 0.200i·25-s + (−0.756 + 0.756i)27-s + (−0.995 + 0.995i)29-s + 1.88i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.177 - 0.984i$
Analytic conductor: \(8.71936\)
Root analytic conductor: \(2.95285\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1),\ -0.177 - 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.255948 + 0.306340i\)
\(L(\frac12)\) \(\approx\) \(0.255948 + 0.306340i\)
\(L(2)\) 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 \( 1 + (1.58 + 1.58i)T \)
good3 \( 1 + (1.91 + 1.91i)T + 9iT^{2} \)
7 \( 1 + 1.82T + 49T^{2} \)
11 \( 1 + (8.81 - 8.81i)T - 121iT^{2} \)
13 \( 1 + (6.87 - 6.87i)T - 169iT^{2} \)
17 \( 1 - 18.7T + 289T^{2} \)
19 \( 1 + (-3.65 - 3.65i)T + 361iT^{2} \)
23 \( 1 - 3.89T + 529T^{2} \)
29 \( 1 + (28.8 - 28.8i)T - 841iT^{2} \)
31 \( 1 - 58.4iT - 961T^{2} \)
37 \( 1 + (18.6 + 18.6i)T + 1.36e3iT^{2} \)
41 \( 1 - 39.0iT - 1.68e3T^{2} \)
43 \( 1 + (-21.4 + 21.4i)T - 1.84e3iT^{2} \)
47 \( 1 - 1.17iT - 2.20e3T^{2} \)
53 \( 1 + (-36.9 - 36.9i)T + 2.80e3iT^{2} \)
59 \( 1 + (-64.8 + 64.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (81.7 - 81.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (49.8 + 49.8i)T + 4.48e3iT^{2} \)
71 \( 1 + 67.8T + 5.04e3T^{2} \)
73 \( 1 + 94.5iT - 5.32e3T^{2} \)
79 \( 1 - 55.2iT - 6.24e3T^{2} \)
83 \( 1 + (51.2 + 51.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 114. iT - 7.92e3T^{2} \)
97 \( 1 + 66.1T + 9.40e3T^{2} \)
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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

−12.01244087171500420247553277251, −10.82945497516696171738286326425, −9.857805460314512848560480215193, −8.883537774996471072985853376702, −7.53366114021587620436217042518, −7.02205718381011719187105134789, −5.73967206704714658689900462710, −4.82442317698003102853201405466, −3.29484885051499800111420630222, −1.49702064956196379256827591276, 0.20747213571372292672666613821, 2.70425396118459207781403376023, 3.96688698279117941541230926775, 5.28965329271262664228553712580, 5.90645553527557754590895222984, 7.45388176557413002429131686211, 8.130800421180426270344599971004, 9.624371567828958346773257644247, 10.28367093956080466434552557109, 11.12062164123460921999046265062

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