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Properties

Label 2-320-320.27-c1-0-1
Degree $2$
Conductor $320$
Sign $-0.456 + 0.889i$
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  + (−0.123 + 1.40i)2-s + (−1.05 + 0.209i)3-s + (−1.96 − 0.347i)4-s + (−0.464 + 2.18i)5-s + (−0.165 − 1.51i)6-s + (0.103 + 0.249i)7-s + (0.732 − 2.73i)8-s + (−1.70 + 0.705i)9-s + (−3.02 − 0.923i)10-s + (−0.477 − 0.715i)11-s + (2.15 − 0.0469i)12-s + (−2.63 − 1.75i)13-s + (−0.364 + 0.114i)14-s + (0.0306 − 2.40i)15-s + (3.75 + 1.36i)16-s − 5.79·17-s + ⋯
L(s)  = 1  + (−0.0871 + 0.996i)2-s + (−0.609 + 0.121i)3-s + (−0.984 − 0.173i)4-s + (−0.207 + 0.978i)5-s + (−0.0676 − 0.617i)6-s + (0.0390 + 0.0943i)7-s + (0.258 − 0.965i)8-s + (−0.567 + 0.235i)9-s + (−0.956 − 0.292i)10-s + (−0.144 − 0.215i)11-s + (0.620 − 0.0135i)12-s + (−0.729 − 0.487i)13-s + (−0.0973 + 0.0307i)14-s + (0.00790 − 0.620i)15-s + (0.939 + 0.342i)16-s − 1.40·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.456 + 0.889i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.456 + 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117147 - 0.191682i\)
\(L(\frac12)\) \(\approx\) \(0.117147 - 0.191682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.123 - 1.40i)T \)
5 \( 1 + (0.464 - 2.18i)T \)
good3 \( 1 + (1.05 - 0.209i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.103 - 0.249i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.477 + 0.715i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (2.63 + 1.75i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + 5.79T + 17T^{2} \)
19 \( 1 + (-6.91 + 1.37i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (5.58 + 2.31i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.531 - 0.794i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + 3.54T + 31T^{2} \)
37 \( 1 + (-1.33 - 1.99i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (6.95 - 2.88i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.46 - 7.34i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 - 9.88iT - 47T^{2} \)
53 \( 1 + (1.96 - 9.88i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (-0.649 + 3.26i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (5.62 - 8.42i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (0.324 + 1.62i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (-4.30 - 1.78i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (7.84 + 3.25i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-3.31 - 3.31i)T + 79iT^{2} \)
83 \( 1 + (5.92 + 3.95i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (3.67 - 8.88i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (8.82 - 8.82i)T - 97iT^{2} \)
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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

−12.14428259327577997238734638325, −11.23520871118410751458816861942, −10.39009535748531898709718762456, −9.443670358467531479788132674892, −8.227623725735684678153149465109, −7.39127049936502982662245521167, −6.41631560268953998391966448572, −5.57383709953868719883136472710, −4.51299379329754192916059445045, −2.92406983574845225989471711949, 0.17262287570985915886238921214, 1.94237668458950629864650801602, 3.69615400740666160987427607259, 4.86250071410275849970680705031, 5.63942126439694098836711534723, 7.28436068393443190472440469730, 8.470162196350236409252856371513, 9.262998934302308598659389585044, 10.10777263719964084189192593179, 11.36098634685490153529953289290

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