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Properties

Label 2-320-20.3-c1-0-3
Degree $2$
Conductor $320$
Sign $0.850 - 0.525i$
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 + (3 + 3i)7-s + i·9-s + 2i·11-s + (−3 − 3i)13-s + (1 + 3i)15-s + (1 − i)17-s + 4·19-s + 6·21-s + (1 − i)23-s + (−3 − 4i)25-s + (4 + 4i)27-s − 10i·31-s + (2 + 2i)33-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (1.13 + 1.13i)7-s + 0.333i·9-s + 0.603i·11-s + (−0.832 − 0.832i)13-s + (0.258 + 0.774i)15-s + (0.242 − 0.242i)17-s + 0.917·19-s + 1.30·21-s + (0.208 − 0.208i)23-s + (−0.600 − 0.800i)25-s + (0.769 + 0.769i)27-s − 1.79i·31-s + (0.348 + 0.348i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51698 + 0.430942i\)
\(L(\frac12)\) \(\approx\) \(1.51698 + 0.430942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (-3 - 3i)T + 47iT^{2} \)
53 \( 1 + (-5 - 5i)T + 53iT^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
97 \( 1 + (3 - 3i)T - 97iT^{2} \)
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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

−11.81591366705771694113443395954, −10.91406864582119726877925109974, −9.863288933263972862940294754371, −8.664646218729736647431524457543, −7.65693164585946708203716146088, −7.41312387154114080138329532482, −5.76657328954933454743612435381, −4.72433740835131608138449855835, −2.92838589410769449520653675639, −2.09253552181524448462198734667, 1.25787883574846595639059045570, 3.42519377028204504433773615563, 4.39210322439505830010682398136, 5.16562771064088594595218800447, 6.96600513175471902469013514923, 7.926287177670478020845990093964, 8.721526712404733872751331346150, 9.574229728358341599138416755870, 10.57734156767100509843644300402, 11.60867118319147795173080593192

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