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Properties

Label 2-320-40.3-c1-0-7
Degree $2$
Conductor $320$
Sign $0.973 + 0.229i$
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  + (1 + i)3-s + (1 − 2i)5-s + (−1 − i)7-s − i·9-s + 4·11-s + (3 − 3i)13-s + (3 − i)15-s + (−3 + 3i)17-s + 6i·19-s − 2i·21-s + (−3 + 3i)23-s + (−3 − 4i)25-s + (4 − 4i)27-s − 2·29-s + 6i·31-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)3-s + (0.447 − 0.894i)5-s + (−0.377 − 0.377i)7-s − 0.333i·9-s + 1.20·11-s + (0.832 − 0.832i)13-s + (0.774 − 0.258i)15-s + (−0.727 + 0.727i)17-s + 1.37i·19-s − 0.436i·21-s + (−0.625 + 0.625i)23-s + (−0.600 − 0.800i)25-s + (0.769 − 0.769i)27-s − 0.371·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67650 - 0.195202i\)
\(L(\frac12)\) \(\approx\) \(1.67650 - 0.195202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (3 - 3i)T - 17iT^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (9 + 9i)T + 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 + (9 - 9i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (3 + 3i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−11.68837005239998595916829500718, −10.33616522897688550485706945452, −9.717172452821239387655322594668, −8.801708660031791190087577861401, −8.203404459831289937590137128669, −6.56410494578771539097241739500, −5.71506287885804007776026299680, −4.16317567156087977885479179183, −3.52281069771939813450087708539, −1.44439507623810985111621831824, 1.93694329587974974996183938080, 2.96927320154170829308493493698, 4.43660479879381727218012065668, 6.21382746456286194991743353501, 6.70417239734258928387200378300, 7.76297179005460585182312075616, 9.071570556925653699035058463326, 9.462901133023270393262549780446, 11.02032927822570671658654264873, 11.42970449629801196089870221042

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