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Properties

Label 2-320-320.3-c1-0-39
Degree $2$
Conductor $320$
Sign $-0.551 - 0.834i$
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  + (−1.16 − 0.808i)2-s + (−1.76 − 2.64i)3-s + (0.693 + 1.87i)4-s + (1.88 − 1.20i)5-s + (−0.0871 + 4.49i)6-s + (−2.23 − 0.926i)7-s + (0.712 − 2.73i)8-s + (−2.71 + 6.56i)9-s + (−3.15 − 0.132i)10-s + (−1.19 − 6.02i)11-s + (3.73 − 5.14i)12-s + (−1.87 − 0.372i)13-s + (1.84 + 2.88i)14-s + (−6.50 − 2.86i)15-s + (−3.03 + 2.60i)16-s − 1.85·17-s + ⋯
L(s)  = 1  + (−0.820 − 0.571i)2-s + (−1.01 − 1.52i)3-s + (0.346 + 0.938i)4-s + (0.843 − 0.536i)5-s + (−0.0355 + 1.83i)6-s + (−0.845 − 0.350i)7-s + (0.251 − 0.967i)8-s + (−0.906 + 2.18i)9-s + (−0.999 − 0.0419i)10-s + (−0.361 − 1.81i)11-s + (1.07 − 1.48i)12-s + (−0.519 − 0.103i)13-s + (0.493 + 0.770i)14-s + (−1.67 − 0.740i)15-s + (−0.759 + 0.650i)16-s − 0.450·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.551 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178712 + 0.332268i\)
\(L(\frac12)\) \(\approx\) \(0.178712 + 0.332268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.808i)T \)
5 \( 1 + (-1.88 + 1.20i)T \)
good3 \( 1 + (1.76 + 2.64i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (2.23 + 0.926i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.19 + 6.02i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (1.87 + 0.372i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 + 1.85T + 17T^{2} \)
19 \( 1 + (-2.78 - 4.17i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-2.16 - 5.21i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.129 - 0.648i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 + (-0.164 - 0.828i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.22 + 2.95i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (4.60 + 3.07i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 - 2.68iT - 47T^{2} \)
53 \( 1 + (-4.23 - 2.83i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (4.11 + 2.74i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (-1.38 + 6.98i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (7.90 - 5.28i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (0.670 + 1.61i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.47 + 8.39i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-7.46 + 7.46i)T - 79iT^{2} \)
83 \( 1 + (3.82 + 0.761i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (2.45 - 1.01i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (0.635 + 0.635i)T + 97iT^{2} \)
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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

−11.08882676487832861410530799172, −10.28357479292400493148524439146, −9.152713085136475380290549393048, −8.116218091676826806273174060911, −7.19967484721772092476885220407, −6.21058744147173932778928483703, −5.46319640785411645637418377989, −3.12467146494477778374659190571, −1.62523920436756827841299105239, −0.38922289141363919920392371830, 2.60984900739332788823359066448, 4.64115032125729901084853017300, 5.37281874078028104157050472659, 6.45216623309411667142797599528, 7.10559721347414767842912139758, 9.112237425555568968020091447052, 9.577534662653020912300629328106, 10.16869977949102865780561307087, 10.82616092305546468014525696833, 11.86236058990823116997249198014

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