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Properties

Label 2-320-80.3-c1-0-0
Degree $2$
Conductor $320$
Sign $-0.700 - 0.714i$
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  − 0.614·3-s + (0.832 + 2.07i)5-s + (−2.83 + 2.83i)7-s − 2.62·9-s + (−1.95 − 1.95i)11-s − 2.05i·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 + 3.45·27-s + (0.835 − 0.835i)29-s + 2.35i·31-s + ⋯
L(s)  = 1  − 0.354·3-s + (0.372 + 0.928i)5-s + (−1.07 + 1.07i)7-s − 0.874·9-s + (−0.590 − 0.590i)11-s − 0.569i·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 + 0.664·27-s + (0.155 − 0.155i)29-s + 0.423i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.700 - 0.714i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.700 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255115 + 0.607456i\)
\(L(\frac12)\) \(\approx\) \(0.255115 + 0.607456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.832 - 2.07i)T \)
good3 \( 1 + 0.614T + 3T^{2} \)
7 \( 1 + (2.83 - 2.83i)T - 7iT^{2} \)
11 \( 1 + (1.95 + 1.95i)T + 11iT^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
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 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 0.849iT - 43T^{2} \)
47 \( 1 + (-2.72 - 2.72i)T + 47iT^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + (-4.16 + 4.16i)T - 59iT^{2} \)
61 \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (4.39 - 4.39i)T - 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
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

−11.85513493384539258087660528210, −11.01057139624062546810712030230, −10.27679082246021850132984761756, −9.183589789655822164742538212669, −8.346925753244220511951535273033, −6.91262724198791762480557557745, −5.97876996464053467884686044779, −5.47530122219425258358875682217, −3.34236075748381529454032459960, −2.56937653557401978924431758040, 0.46224072978959436469265938679, 2.63278330827213991211527734510, 4.28226765536515779313517652558, 5.20106822630541065439393663806, 6.44508369895175315276962852919, 7.24988786979117555607953886705, 8.664636476217248339552067710700, 9.417488751555539833025705556458, 10.32412753737985828203547275120, 11.26784175479138609607803156180

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