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Properties

Label 2-320-320.243-c1-0-33
Degree $2$
Conductor $320$
Sign $0.905 + 0.424i$
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  + (1.37 + 0.342i)2-s + (0.462 − 2.32i)3-s + (1.76 + 0.939i)4-s + (−0.624 + 2.14i)5-s + (1.43 − 3.03i)6-s + (0.837 − 2.02i)7-s + (2.10 + 1.89i)8-s + (−2.43 − 1.00i)9-s + (−1.59 + 2.73i)10-s + (2.81 + 1.88i)11-s + (3.00 − 3.67i)12-s + (−3.21 − 4.80i)13-s + (1.84 − 2.48i)14-s + (4.70 + 2.44i)15-s + (2.23 + 3.31i)16-s − 0.558·17-s + ⋯
L(s)  = 1  + (0.970 + 0.242i)2-s + (0.267 − 1.34i)3-s + (0.882 + 0.469i)4-s + (−0.279 + 0.960i)5-s + (0.584 − 1.23i)6-s + (0.316 − 0.764i)7-s + (0.742 + 0.669i)8-s + (−0.810 − 0.335i)9-s + (−0.503 + 0.864i)10-s + (0.849 + 0.567i)11-s + (0.867 − 1.06i)12-s + (−0.891 − 1.33i)13-s + (0.492 − 0.664i)14-s + (1.21 + 0.631i)15-s + (0.558 + 0.829i)16-s − 0.135·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.905 + 0.424i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.905 + 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40142 - 0.534804i\)
\(L(\frac12)\) \(\approx\) \(2.40142 - 0.534804i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.342i)T \)
5 \( 1 + (0.624 - 2.14i)T \)
good3 \( 1 + (-0.462 + 2.32i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (-0.837 + 2.02i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.81 - 1.88i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (3.21 + 4.80i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + 0.558T + 17T^{2} \)
19 \( 1 + (0.234 - 1.17i)T + (-17.5 - 7.27i)T^{2} \)
23 \( 1 + (7.72 - 3.19i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (5.71 - 3.81i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 - 5.46T + 31T^{2} \)
37 \( 1 + (0.191 + 0.127i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-7.38 - 3.05i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (6.63 - 1.31i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 - 0.739iT - 47T^{2} \)
53 \( 1 + (4.60 - 0.916i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (-0.580 + 0.115i)T + (54.5 - 22.5i)T^{2} \)
61 \( 1 + (11.3 - 7.58i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (2.84 + 0.566i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-5.95 + 2.46i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-8.59 + 3.55i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.94 + 9.94i)T - 79iT^{2} \)
83 \( 1 + (-4.22 - 6.32i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (5.44 + 13.1i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (4.42 + 4.42i)T + 97iT^{2} \)
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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

−11.97567201099014792745233871091, −10.93718030014911915937379681369, −9.967290291578350842088168187647, −7.88838574965001642159537706659, −7.63861448810742434315920276306, −6.81046592829494561949701139325, −5.92733382650978322144332877608, −4.33006776973286938664336019396, −3.10908134106039347515262475349, −1.83789290118980685722576191276, 2.15073643515314888492786518334, 3.85160234203413291386301216883, 4.42760456556539120760576423072, 5.30771204694872967758495878125, 6.46425637398198089683789041468, 8.102687664527255346337422517049, 9.220145930780375833340691376849, 9.693985551026871264492983848981, 11.00414149496419670945071923100, 11.86529208348690477491571294458

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