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Properties

Label 2-320-320.107-c1-0-13
Degree $2$
Conductor $320$
Sign $-0.264 - 0.964i$
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  + (−0.332 + 1.37i)2-s + (0.678 − 1.01i)3-s + (−1.77 − 0.913i)4-s + (1.34 + 1.78i)5-s + (1.17 + 1.27i)6-s + (−2.60 + 1.07i)7-s + (1.84 − 2.14i)8-s + (0.576 + 1.39i)9-s + (−2.90 + 1.25i)10-s + (−0.617 + 3.10i)11-s + (−2.13 + 1.18i)12-s + (0.324 − 0.0646i)13-s + (−0.617 − 3.94i)14-s + (2.72 − 0.156i)15-s + (2.33 + 3.25i)16-s + 7.28·17-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)2-s + (0.391 − 0.586i)3-s + (−0.889 − 0.456i)4-s + (0.602 + 0.798i)5-s + (0.477 + 0.518i)6-s + (−0.985 + 0.408i)7-s + (0.653 − 0.757i)8-s + (0.192 + 0.464i)9-s + (−0.917 + 0.397i)10-s + (−0.186 + 0.935i)11-s + (−0.616 + 0.342i)12-s + (0.0901 − 0.0179i)13-s + (−0.165 − 1.05i)14-s + (0.704 − 0.0404i)15-s + (0.582 + 0.812i)16-s + 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.264 - 0.964i$
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: \((2,\ 320,\ (\ :1/2),\ -0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.725444 + 0.950994i\)
\(L(\frac12)\) \(\approx\) \(0.725444 + 0.950994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.332 - 1.37i)T \)
5 \( 1 + (-1.34 - 1.78i)T \)
good3 \( 1 + (-0.678 + 1.01i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (2.60 - 1.07i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.617 - 3.10i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-0.324 + 0.0646i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 - 7.28T + 17T^{2} \)
19 \( 1 + (1.21 - 1.81i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.57 - 3.79i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.750 + 3.77i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 5.19T + 31T^{2} \)
37 \( 1 + (-1.44 + 7.25i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (1.84 + 4.45i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.87 + 3.92i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + 7.32iT - 47T^{2} \)
53 \( 1 + (3.02 - 2.01i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (2.56 - 1.71i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.77 - 8.91i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-5.56 - 3.71i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (-5.26 + 12.7i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.67 + 8.86i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (3.96 + 3.96i)T + 79iT^{2} \)
83 \( 1 + (-8.57 + 1.70i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (3.24 + 1.34i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-9.04 + 9.04i)T - 97iT^{2} \)
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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

−12.29045414294099971650113576245, −10.50301827844836677207316767129, −9.899357705487676490819444346450, −9.106023215884117663411804410980, −7.66327090450075320896372084475, −7.32023221276867845006085040088, −6.15992871985375724237484829284, −5.40216352890398569276459020084, −3.58724449301564356994080795207, −2.00902457092343181033748458134, 0.968437827832516850713086704980, 2.97889066954528434714461185483, 3.80658422137465401355724770995, 5.06157942890774931417951566291, 6.31663734105288907176546801776, 8.017698293965463230050709766505, 8.898615933312782243239648552978, 9.714547437335704947866206529680, 10.10759320654359691812014890372, 11.21455952444584596490244676930

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