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Properties

Label 2-320-320.43-c1-0-37
Degree $2$
Conductor $320$
Sign $-0.587 + 0.809i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.126 − 1.40i)2-s + (0.572 + 0.382i)3-s + (−1.96 − 0.355i)4-s + (0.0653 − 2.23i)5-s + (0.611 − 0.758i)6-s + (1.18 + 2.86i)7-s + (−0.748 + 2.72i)8-s + (−0.966 − 2.33i)9-s + (−3.14 − 0.373i)10-s + (1.08 − 5.45i)11-s + (−0.991 − 0.956i)12-s + (−1.30 − 6.54i)13-s + (4.18 − 1.31i)14-s + (0.892 − 1.25i)15-s + (3.74 + 1.39i)16-s + 3.01i·17-s + ⋯
L(s)  = 1  + (0.0891 − 0.996i)2-s + (0.330 + 0.220i)3-s + (−0.984 − 0.177i)4-s + (0.0292 − 0.999i)5-s + (0.249 − 0.309i)6-s + (0.448 + 1.08i)7-s + (−0.264 + 0.964i)8-s + (−0.322 − 0.777i)9-s + (−0.992 − 0.118i)10-s + (0.327 − 1.64i)11-s + (−0.286 − 0.276i)12-s + (−0.360 − 1.81i)13-s + (1.11 − 0.350i)14-s + (0.230 − 0.324i)15-s + (0.936 + 0.349i)16-s + 0.730i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603565 - 1.18319i\)
\(L(\frac12)\) \(\approx\) \(0.603565 - 1.18319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.126 + 1.40i)T \)
5 \( 1 + (-0.0653 + 2.23i)T \)
good3 \( 1 + (-0.572 - 0.382i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-1.18 - 2.86i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.08 + 5.45i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.30 + 6.54i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 - 3.01iT - 17T^{2} \)
19 \( 1 + (1.01 - 1.51i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-4.46 - 1.85i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.61 - 8.11i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 4.36T + 31T^{2} \)
37 \( 1 + (-4.45 - 0.885i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.716 + 1.73i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.482 + 0.722i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 + (-2.33 - 3.50i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-3.85 + 2.57i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.36 - 6.88i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.14 - 1.72i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (1.09 - 2.64i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.804 - 0.333i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-11.1 - 11.1i)T + 79iT^{2} \)
83 \( 1 + (-1.37 - 6.91i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (2.21 + 0.917i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (6.21 + 6.21i)T + 97iT^{2} \)
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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

−11.37127155029088574530332763291, −10.50455386386147243539502533209, −9.259271703230511183352714549818, −8.669447649940278303857124006412, −8.200294509125093683630560243044, −5.71805827966108480227337116088, −5.39803948309960291977945958514, −3.77125776103842408886746973218, −2.82719693258141792921720403297, −0.976933761809875375727601088156, 2.23663608902334198580867890336, 4.13078836427929117761057668248, 4.81102622420863804191943025761, 6.56194608910639414278220919913, 7.23309264886673267186214405245, 7.64753998756532308099448844405, 9.107565343658019297761901641377, 9.892795883112424210751700415609, 10.96982827399740400211393684747, 11.97088743444571176748252748678

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