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Properties

Label 2-320-320.243-c1-0-16
Degree $2$
Conductor $320$
Sign $-0.967 - 0.251i$
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  + (0.640 + 1.26i)2-s + (−0.468 + 2.35i)3-s + (−1.17 + 1.61i)4-s + (2.11 − 0.715i)5-s + (−3.27 + 0.918i)6-s + (−0.666 + 1.60i)7-s + (−2.79 − 0.452i)8-s + (−2.56 − 1.06i)9-s + (2.25 + 2.21i)10-s + (3.36 + 2.24i)11-s + (−3.25 − 3.53i)12-s + (−1.02 − 1.52i)13-s + (−2.45 + 0.190i)14-s + (0.694 + 5.32i)15-s + (−1.21 − 3.81i)16-s − 5.10·17-s + ⋯
L(s)  = 1  + (0.452 + 0.891i)2-s + (−0.270 + 1.36i)3-s + (−0.589 + 0.807i)4-s + (0.947 − 0.320i)5-s + (−1.33 + 0.375i)6-s + (−0.251 + 0.607i)7-s + (−0.987 − 0.160i)8-s + (−0.855 − 0.354i)9-s + (0.714 + 0.699i)10-s + (1.01 + 0.678i)11-s + (−0.939 − 1.02i)12-s + (−0.282 − 0.423i)13-s + (−0.656 + 0.0508i)14-s + (0.179 + 1.37i)15-s + (−0.304 − 0.952i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.967 - 0.251i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.967 - 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198275 + 1.54838i\)
\(L(\frac12)\) \(\approx\) \(0.198275 + 1.54838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.640 - 1.26i)T \)
5 \( 1 + (-2.11 + 0.715i)T \)
good3 \( 1 + (0.468 - 2.35i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (0.666 - 1.60i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-3.36 - 2.24i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (1.02 + 1.52i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + 5.10T + 17T^{2} \)
19 \( 1 + (-0.766 + 3.85i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (-4.44 + 1.84i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.75 + 1.84i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 - 4.32T + 31T^{2} \)
37 \( 1 + (-0.976 - 0.652i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.800 - 0.331i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (12.5 - 2.50i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 - 13.1iT - 47T^{2} \)
53 \( 1 + (-7.12 + 1.41i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (4.98 - 0.990i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (5.29 - 3.53i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-5.63 - 1.12i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-5.58 + 2.31i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-14.4 + 5.99i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.413 + 0.413i)T - 79iT^{2} \)
83 \( 1 + (7.33 + 10.9i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-0.256 - 0.618i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (-0.386 - 0.386i)T + 97iT^{2} \)
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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

−12.20474434768072187633142643013, −11.08647460853806602838794409086, −9.821407046011197462572767231868, −9.295342063990826548536095753708, −8.608541978453059240381842976376, −6.86860313083187869575197635850, −6.07023054930613555845708213639, −4.90534031554452846079509549585, −4.47611090174364217521024940335, −2.84989140068551213947454902946, 1.12425402240193999205462826884, 2.22388206678496500084259445769, 3.69608595725156234008561935661, 5.27907999436846025839056741345, 6.54522323709997171273540332560, 6.75635045564683450693728995539, 8.529138464736400774948227927706, 9.529355406356739995594729895717, 10.49098596048970771359720581598, 11.43843546970470078233247289492

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