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Properties

Label 2-320-20.19-c8-0-25
Degree $2$
Conductor $320$
Sign $-0.620 - 0.783i$
Analytic cond. $130.361$
Root an. cond. $11.4175$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 125.·3-s + (−388. − 489. i)5-s + 4.12e3·7-s + 9.20e3·9-s + 2.69e4i·11-s − 8.36e3i·13-s + (4.87e4 + 6.15e4i)15-s + 7.21e4i·17-s + 1.37e5i·19-s − 5.18e5·21-s − 4.62e4·23-s + (−8.93e4 + 3.80e5i)25-s − 3.31e5·27-s + 9.75e5·29-s + 8.09e5i·31-s + ⋯
L(s)  = 1  − 1.55·3-s + (−0.620 − 0.783i)5-s + 1.71·7-s + 1.40·9-s + 1.83i·11-s − 0.292i·13-s + (0.962 + 1.21i)15-s + 0.864i·17-s + 1.05i·19-s − 2.66·21-s − 0.165·23-s + (−0.228 + 0.973i)25-s − 0.623·27-s + 1.37·29-s + 0.876i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.620 - 0.783i$
Analytic conductor: \(130.361\)
Root analytic conductor: \(11.4175\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :4),\ -0.620 - 0.783i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.8866237813\)
\(L(\frac12)\) \(\approx\) \(0.8866237813\)
\(L(5)\) 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 \( 1 + (388. + 489. i)T \)
good3 \( 1 + 125.T + 6.56e3T^{2} \)
7 \( 1 - 4.12e3T + 5.76e6T^{2} \)
11 \( 1 - 2.69e4iT - 2.14e8T^{2} \)
13 \( 1 + 8.36e3iT - 8.15e8T^{2} \)
17 \( 1 - 7.21e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.37e5iT - 1.69e10T^{2} \)
23 \( 1 + 4.62e4T + 7.83e10T^{2} \)
29 \( 1 - 9.75e5T + 5.00e11T^{2} \)
31 \( 1 - 8.09e5iT - 8.52e11T^{2} \)
37 \( 1 - 1.67e6iT - 3.51e12T^{2} \)
41 \( 1 + 2.35e6T + 7.98e12T^{2} \)
43 \( 1 + 2.52e5T + 1.16e13T^{2} \)
47 \( 1 - 5.36e6T + 2.38e13T^{2} \)
53 \( 1 + 7.99e5iT - 6.22e13T^{2} \)
59 \( 1 + 1.84e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.40e7T + 1.91e14T^{2} \)
67 \( 1 - 1.89e7T + 4.06e14T^{2} \)
71 \( 1 + 2.68e7iT - 6.45e14T^{2} \)
73 \( 1 - 4.89e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.02e7iT - 1.51e15T^{2} \)
83 \( 1 - 8.40e6T + 2.25e15T^{2} \)
89 \( 1 + 5.91e7T + 3.93e15T^{2} \)
97 \( 1 + 4.66e7iT - 7.83e15T^{2} \)
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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

−10.69833850200098368800940372780, −10.00691372472865525569606424239, −8.443553872452914513019163049360, −7.76754264839814847178176961019, −6.70426787950968399084198674751, −5.37459085577047871625971112957, −4.81673750266596017820417314308, −4.16634097397969868256354475187, −1.74043424334608356367218804622, −1.13934112748768436806582103693, 0.29427969702345009654817351195, 1.01220733928332028626516880387, 2.67000696149184757213519896352, 4.18501643544171351287284998309, 5.06657842622675155288350518201, 5.94002267173785179798752305150, 6.91768188651357926289086672711, 7.87344610164907141557484942630, 8.833868956657634959308887925268, 10.55853159295520352605629045978

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