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Properties

Label 2-320-320.227-c1-0-28
Degree $2$
Conductor $320$
Sign $0.716 + 0.697i$
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  + (1.18 − 0.777i)2-s + (−0.0766 − 0.114i)3-s + (0.791 − 1.83i)4-s + (0.840 + 2.07i)5-s + (−0.179 − 0.0759i)6-s + (−0.433 + 1.04i)7-s + (−0.493 − 2.78i)8-s + (1.14 − 2.75i)9-s + (2.60 + 1.79i)10-s + (6.05 − 1.20i)11-s + (−0.271 + 0.0500i)12-s + (−4.32 − 0.860i)13-s + (0.301 + 1.57i)14-s + (0.173 − 0.255i)15-s + (−2.74 − 2.90i)16-s + 6.12i·17-s + ⋯
L(s)  = 1  + (0.835 − 0.549i)2-s + (−0.0442 − 0.0662i)3-s + (0.395 − 0.918i)4-s + (0.376 + 0.926i)5-s + (−0.0734 − 0.0310i)6-s + (−0.164 + 0.395i)7-s + (−0.174 − 0.984i)8-s + (0.380 − 0.918i)9-s + (0.823 + 0.567i)10-s + (1.82 − 0.363i)11-s + (−0.0783 + 0.0144i)12-s + (−1.19 − 0.238i)13-s + (0.0806 + 0.420i)14-s + (0.0447 − 0.0659i)15-s + (−0.686 − 0.726i)16-s + 1.48i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02838 - 0.824351i\)
\(L(\frac12)\) \(\approx\) \(2.02838 - 0.824351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.777i)T \)
5 \( 1 + (-0.840 - 2.07i)T \)
good3 \( 1 + (0.0766 + 0.114i)T + (-1.14 + 2.77i)T^{2} \)
7 \( 1 + (0.433 - 1.04i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-6.05 + 1.20i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (4.32 + 0.860i)T + (12.0 + 4.97i)T^{2} \)
17 \( 1 - 6.12iT - 17T^{2} \)
19 \( 1 + (-1.99 + 1.33i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-2.05 + 0.850i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (8.76 + 1.74i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + 7.25T + 31T^{2} \)
37 \( 1 + (-0.816 - 4.10i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (2.16 - 5.23i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.215 + 0.144i)T + (16.4 + 39.7i)T^{2} \)
47 \( 1 - 5.40T + 47T^{2} \)
53 \( 1 + (5.02 + 3.35i)T + (20.2 + 48.9i)T^{2} \)
59 \( 1 + (-0.444 + 0.665i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (2.30 + 0.457i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (10.1 - 6.76i)T + (25.6 - 61.8i)T^{2} \)
71 \( 1 + (-1.77 - 4.29i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-9.20 + 3.81i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.16 + 9.16i)T - 79iT^{2} \)
83 \( 1 + (4.41 + 0.878i)T + (76.6 + 31.7i)T^{2} \)
89 \( 1 + (4.31 - 1.78i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (2.41 - 2.41i)T - 97iT^{2} \)
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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

−11.64461071588105480487967658078, −10.81149231670064837367682700432, −9.688254369586582784465493820045, −9.214054498196756736876690295926, −7.22329963045287258759638462799, −6.43533004785009913096374949937, −5.70683618370538790267653245216, −4.06207707085721478172309150448, −3.21521871486763999060244594113, −1.70479414518798112283487882851, 1.96527966403894897617661413192, 3.84553263824715379731613365883, 4.79357092770689268181428673835, 5.56682768193905179121946584063, 7.07199509252977323519298631071, 7.47229989747272396391754813517, 9.074844150795523405308858026661, 9.570080908422102440032768001579, 11.11881474940069298175834398281, 12.05862832016684168819023388630

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