Properties

Label 2-320-80.67-c1-0-7
Degree $2$
Conductor $320$
Sign $-0.435 + 0.900i$
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  + 0.614i·3-s + (−2.07 − 0.832i)5-s + (−2.83 − 2.83i)7-s + 2.62·9-s + (−1.95 − 1.95i)11-s − 2.05·13-s + (0.511 − 1.27i)15-s + (−4.06 − 4.06i)17-s + (−0.683 − 0.683i)19-s + (1.74 − 1.74i)21-s + (4.95 − 4.95i)23-s + (3.61 + 3.45i)25-s + 3.45i·27-s + (−0.835 + 0.835i)29-s + 2.35i·31-s + ⋯
L(s)  = 1  + 0.354i·3-s + (−0.928 − 0.372i)5-s + (−1.07 − 1.07i)7-s + 0.874·9-s + (−0.590 − 0.590i)11-s − 0.569·13-s + (0.132 − 0.329i)15-s + (−0.986 − 0.986i)17-s + (−0.156 − 0.156i)19-s + (0.380 − 0.380i)21-s + (1.03 − 1.03i)23-s + (0.723 + 0.690i)25-s + 0.664i·27-s + (−0.155 + 0.155i)29-s + 0.423i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.435 + 0.900i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338220 - 0.539341i\)
\(L(\frac12)\) \(\approx\) \(0.338220 - 0.539341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.832i)T \)
good3 \( 1 - 0.614iT - 3T^{2} \)
7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 + 1.95i)T + 11iT^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 + (4.06 + 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 + 0.683i)T + 19iT^{2} \)
23 \( 1 + (-4.95 + 4.95i)T - 23iT^{2} \)
29 \( 1 + (0.835 - 0.835i)T - 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 0.849T + 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 + 5.17iT - 53T^{2} \)
59 \( 1 + (4.16 - 4.16i)T - 59iT^{2} \)
61 \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \)
67 \( 1 - 1.73T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−11.12024103024853147349405159430, −10.48860277650012428873978429903, −9.526187984472204098704491253376, −8.578971405539956914182629695335, −7.24228495885408250908358477422, −6.85665017698941788568396849440, −4.99877987162768082388792575939, −4.16421934849106456722099286198, −3.06396763103394117051343131837, −0.44246839303484245370774739101, 2.27359684998385539131593958526, 3.57029547468914458182975386746, 4.87141560751314894430820821807, 6.31436501392652795228675174019, 7.10047220735167409305277624809, 8.009965546303252980157670750983, 9.169027773905613756202650599729, 10.04353019605462829494186651489, 11.06829070340525553031733263194, 12.15393820380459105704886883180

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