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Properties

Label 2-320-320.107-c1-0-1
Degree $2$
Conductor $320$
Sign $-0.985 - 0.167i$
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  + (1.01 − 0.988i)2-s + (−1.81 + 2.72i)3-s + (0.0452 − 1.99i)4-s + (−2.03 + 0.923i)5-s + (0.851 + 4.55i)6-s + (−0.386 + 0.160i)7-s + (−1.93 − 2.06i)8-s + (−2.95 − 7.12i)9-s + (−1.14 + 2.94i)10-s + (0.0859 − 0.432i)11-s + (5.35 + 3.75i)12-s + (−3.24 + 0.645i)13-s + (−0.232 + 0.544i)14-s + (1.19 − 7.22i)15-s + (−3.99 − 0.180i)16-s − 5.57·17-s + ⋯
L(s)  = 1  + (0.715 − 0.699i)2-s + (−1.05 + 1.57i)3-s + (0.0226 − 0.999i)4-s + (−0.910 + 0.412i)5-s + (0.347 + 1.85i)6-s + (−0.146 + 0.0605i)7-s + (−0.682 − 0.730i)8-s + (−0.984 − 2.37i)9-s + (−0.362 + 0.931i)10-s + (0.0259 − 0.130i)11-s + (1.54 + 1.08i)12-s + (−0.900 + 0.179i)13-s + (−0.0621 + 0.145i)14-s + (0.307 − 1.86i)15-s + (−0.998 − 0.0452i)16-s − 1.35·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.985 - 0.167i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.985 - 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0154664 + 0.182924i\)
\(L(\frac12)\) \(\approx\) \(0.0154664 + 0.182924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.01 + 0.988i)T \)
5 \( 1 + (2.03 - 0.923i)T \)
good3 \( 1 + (1.81 - 2.72i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.386 - 0.160i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.0859 + 0.432i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (3.24 - 0.645i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + 5.57T + 17T^{2} \)
19 \( 1 + (2.10 - 3.14i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.02 - 2.47i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.512 + 2.57i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 + (2.03 - 10.2i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (2.59 + 6.26i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.81 + 1.21i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 - 7.69iT - 47T^{2} \)
53 \( 1 + (-7.26 + 4.85i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (1.81 - 1.21i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.32 - 6.68i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (5.40 + 3.61i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (3.18 - 7.69i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.77 + 4.28i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.40 + 2.40i)T + 79iT^{2} \)
83 \( 1 + (8.73 - 1.73i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (14.4 + 5.98i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-12.2 + 12.2i)T - 97iT^{2} \)
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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

−11.74131642039153574357674240587, −11.29494869524902576396060617764, −10.37060629409884675812339651694, −9.852848173767096503972749308429, −8.695508802131453164788395825186, −6.79032746555572447678600905109, −5.88051060770833865516387177889, −4.64013945270902107122904814085, −4.18008088770862535765554909691, −2.99710592032956925568585890007, 0.11121879265853175736667452237, 2.44873998108587389325185907409, 4.44446539808179306011280952922, 5.27337334098974584859854952250, 6.54625335913786396040983660412, 7.03914152978853807883462120167, 7.912834912346584488510153183019, 8.758776550152215564522878802718, 10.83450938593108397248864425504, 11.65908102846094693811648616346

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