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Properties

Label 2-320-5.4-c3-0-24
Degree $2$
Conductor $320$
Sign $-0.786 + 0.617i$
Analytic cond. $18.8806$
Root an. cond. $4.34518$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.89i·3-s + (−8.79 + 6.89i)5-s + 16.6i·7-s + 18.5·9-s − 19.1·11-s − 61.7i·13-s + (20 + 25.5i)15-s + 30.3i·17-s − 59.1·19-s + 48.4·21-s − 205. i·23-s + (29.8 − 121. i)25-s − 132. i·27-s + 8.38·29-s − 331.·31-s + ⋯
L(s)  = 1  − 0.557i·3-s + (−0.786 + 0.617i)5-s + 0.901i·7-s + 0.688·9-s − 0.526·11-s − 1.31i·13-s + (0.344 + 0.439i)15-s + 0.433i·17-s − 0.714·19-s + 0.502·21-s − 1.86i·23-s + (0.238 − 0.971i)25-s − 0.942i·27-s + 0.0536·29-s − 1.91·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(18.8806\)
Root analytic conductor: \(4.34518\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :3/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5580702732\)
\(L(\frac12)\) \(\approx\) \(0.5580702732\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.79 - 6.89i)T \)
good3 \( 1 + 2.89iT - 27T^{2} \)
7 \( 1 - 16.6iT - 343T^{2} \)
11 \( 1 + 19.1T + 1.33e3T^{2} \)
13 \( 1 + 61.7iT - 2.19e3T^{2} \)
17 \( 1 - 30.3iT - 4.91e3T^{2} \)
19 \( 1 + 59.1T + 6.85e3T^{2} \)
23 \( 1 + 205. iT - 1.21e4T^{2} \)
29 \( 1 - 8.38T + 2.43e4T^{2} \)
31 \( 1 + 331.T + 2.97e4T^{2} \)
37 \( 1 - 266. iT - 5.06e4T^{2} \)
41 \( 1 + 320.T + 6.89e4T^{2} \)
43 \( 1 - 83.1iT - 7.95e4T^{2} \)
47 \( 1 + 276. iT - 1.03e5T^{2} \)
53 \( 1 + 390. iT - 1.48e5T^{2} \)
59 \( 1 + 779.T + 2.05e5T^{2} \)
61 \( 1 - 483.T + 2.26e5T^{2} \)
67 \( 1 + 123. iT - 3.00e5T^{2} \)
71 \( 1 + 187.T + 3.57e5T^{2} \)
73 \( 1 + 778. iT - 3.89e5T^{2} \)
79 \( 1 + 446.T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3iT - 5.71e5T^{2} \)
89 \( 1 - 94.8T + 7.04e5T^{2} \)
97 \( 1 + 252. iT - 9.12e5T^{2} \)
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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

−10.73112679114653548231350028569, −10.20841369733421894707597657537, −8.617898684605485884909775164953, −7.993804069761425002324676440210, −7.00037128335449942658688274787, −6.06547048737910821875137169749, −4.75449732035462895812268910544, −3.32899591199924708878378369414, −2.15208588929754242765720112775, −0.20215456829185389233580846427, 1.53547135333477726992629414370, 3.70128110863634133288289175851, 4.29461807831412531131882608339, 5.34303086105939052805173732886, 7.08292252227997729672897248001, 7.56917682204094075861081765214, 8.944712882810944899565149033503, 9.613668568360209249250866877527, 10.74552294894572722104443167816, 11.39064035625470970118755397208

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