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Properties

Label 2-320-20.7-c3-0-7
Degree $2$
Conductor $320$
Sign $-0.999 + 0.00192i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.07 + 6.07i)3-s + (−5.85 + 9.52i)5-s + (−18.8 + 18.8i)7-s + 46.8i·9-s + 21.3i·11-s + (53.7 − 53.7i)13-s + (−93.4 + 22.2i)15-s + (−39.3 − 39.3i)17-s − 16.0·19-s − 228.·21-s + (−1.84 − 1.84i)23-s + (−56.3 − 111. i)25-s + (−120. + 120. i)27-s + 108. i·29-s − 11.4i·31-s + ⋯
L(s)  = 1  + (1.16 + 1.16i)3-s + (−0.524 + 0.851i)5-s + (−1.01 + 1.01i)7-s + 1.73i·9-s + 0.585i·11-s + (1.14 − 1.14i)13-s + (−1.60 + 0.382i)15-s + (−0.561 − 0.561i)17-s − 0.193·19-s − 2.37·21-s + (−0.0167 − 0.0167i)23-s + (−0.450 − 0.892i)25-s + (−0.857 + 0.857i)27-s + 0.697i·29-s − 0.0661i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00192i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.999 + 0.00192i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ -0.999 + 0.00192i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.758692675\)
\(L(\frac12)\) \(\approx\) \(1.758692675\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.85 - 9.52i)T \)
good3 \( 1 + (-6.07 - 6.07i)T + 27iT^{2} \)
7 \( 1 + (18.8 - 18.8i)T - 343iT^{2} \)
11 \( 1 - 21.3iT - 1.33e3T^{2} \)
13 \( 1 + (-53.7 + 53.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (39.3 + 39.3i)T + 4.91e3iT^{2} \)
19 \( 1 + 16.0T + 6.85e3T^{2} \)
23 \( 1 + (1.84 + 1.84i)T + 1.21e4iT^{2} \)
29 \( 1 - 108. iT - 2.43e4T^{2} \)
31 \( 1 + 11.4iT - 2.97e4T^{2} \)
37 \( 1 + (-76.0 - 76.0i)T + 5.06e4iT^{2} \)
41 \( 1 + 305.T + 6.89e4T^{2} \)
43 \( 1 + (-238. - 238. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-93.0 + 93.0i)T - 1.03e5iT^{2} \)
53 \( 1 + (286. - 286. i)T - 1.48e5iT^{2} \)
59 \( 1 - 674.T + 2.05e5T^{2} \)
61 \( 1 - 128.T + 2.26e5T^{2} \)
67 \( 1 + (595. - 595. i)T - 3.00e5iT^{2} \)
71 \( 1 - 1.07e3iT - 3.57e5T^{2} \)
73 \( 1 + (633. - 633. i)T - 3.89e5iT^{2} \)
79 \( 1 - 252.T + 4.93e5T^{2} \)
83 \( 1 + (-12.2 - 12.2i)T + 5.71e5iT^{2} \)
89 \( 1 + 935. iT - 7.04e5T^{2} \)
97 \( 1 + (-12.8 - 12.8i)T + 9.12e5iT^{2} \)
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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

−11.44744289705270462342467515361, −10.46619912677205233346915438443, −9.809729495953514688358282790800, −8.883850600677947813964357733176, −8.201911365635040486536312947547, −6.93016951736134073005804914905, −5.69918193433077891165717444844, −4.23471611869742264581036570329, −3.20482620772581710574459551845, −2.64184666566899165736869662116, 0.54308507034914490615826266741, 1.75053897643511790160273116311, 3.42444405478053393249690964318, 4.13407222087269102282726103348, 6.21763487615123797628619194465, 6.95809089917700125771567704413, 7.963139400311531441192253682191, 8.718970704420605604425357859960, 9.380891809543407248361099608667, 10.81096623592203984210963069164

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