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Properties

Label 2-320-20.7-c1-0-6
Degree $2$
Conductor $320$
Sign $0.0898 + 0.995i$
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  + (−2 − i)5-s − 3i·9-s + (5 − 5i)13-s + (−5 − 5i)17-s + (3 + 4i)25-s + 4i·29-s + (−5 − 5i)37-s + 8·41-s + (−3 + 6i)45-s + 7i·49-s + (−5 + 5i)53-s + 12·61-s + (−15 + 5i)65-s + (5 − 5i)73-s − 9·81-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)5-s − i·9-s + (1.38 − 1.38i)13-s + (−1.21 − 1.21i)17-s + (0.600 + 0.800i)25-s + 0.742i·29-s + (−0.821 − 0.821i)37-s + 1.24·41-s + (−0.447 + 0.894i)45-s + i·49-s + (−0.686 + 0.686i)53-s + 1.53·61-s + (−1.86 + 0.620i)65-s + (0.585 − 0.585i)73-s − 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.0898 + 0.995i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748467 - 0.684014i\)
\(L(\frac12)\) \(\approx\) \(0.748467 - 0.684014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2 + i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-5 + 5i)T - 13iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−11.34802083647064143476429057640, −10.75088767626022843866068533848, −9.290739722645913941443564781581, −8.674534544629272867030782860446, −7.65282297564612043595752202219, −6.59336922864393581770584593663, −5.40503843578542021326014796610, −4.12396539936214544141183389183, −3.12558842572288379734781975579, −0.76085770125354409810161120625, 2.02547983030528401194618204137, 3.73628229388653838275156087841, 4.55279357045582279894781279974, 6.17080292838263670808391925213, 7.00036321406455903231163002005, 8.203325745555624988059499181647, 8.753983348100711241549479155291, 10.21521912329344702226891179652, 11.17739061108679690381095781286, 11.45823637494045206387330414327

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