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Properties

Label 2-320-320.107-c1-0-0
Degree $2$
Conductor $320$
Sign $-0.439 + 0.898i$
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  + (−1.40 − 0.180i)2-s + (−1.12 + 1.68i)3-s + (1.93 + 0.507i)4-s + (−1.49 + 1.66i)5-s + (1.88 − 2.16i)6-s + (−3.56 + 1.47i)7-s + (−2.62 − 1.06i)8-s + (−0.430 − 1.03i)9-s + (2.39 − 2.06i)10-s + (0.785 − 3.94i)11-s + (−3.03 + 2.69i)12-s + (−1.85 + 0.368i)13-s + (5.27 − 1.42i)14-s + (−1.13 − 4.39i)15-s + (3.48 + 1.96i)16-s + 5.96·17-s + ⋯
L(s)  = 1  + (−0.991 − 0.127i)2-s + (−0.651 + 0.974i)3-s + (0.967 + 0.253i)4-s + (−0.666 + 0.745i)5-s + (0.770 − 0.883i)6-s + (−1.34 + 0.558i)7-s + (−0.926 − 0.375i)8-s + (−0.143 − 0.346i)9-s + (0.756 − 0.653i)10-s + (0.236 − 1.19i)11-s + (−0.877 + 0.777i)12-s + (−0.513 + 0.102i)13-s + (1.40 − 0.381i)14-s + (−0.291 − 1.13i)15-s + (0.871 + 0.491i)16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.439 + 0.898i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0177236 - 0.0284197i\)
\(L(\frac12)\) \(\approx\) \(0.0177236 - 0.0284197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.180i)T \)
5 \( 1 + (1.49 - 1.66i)T \)
good3 \( 1 + (1.12 - 1.68i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (3.56 - 1.47i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.785 + 3.94i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.85 - 0.368i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 - 5.96T + 17T^{2} \)
19 \( 1 + (2.30 - 3.45i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (-3.36 + 8.11i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.602 - 3.02i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 9.61T + 31T^{2} \)
37 \( 1 + (-0.220 + 1.10i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (3.18 + 7.69i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.79 + 1.20i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 - 3.37iT - 47T^{2} \)
53 \( 1 + (7.85 - 5.24i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (10.8 - 7.26i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.837 - 4.21i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (1.52 + 1.02i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (-1.14 + 2.77i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.789 + 1.90i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-2.72 - 2.72i)T + 79iT^{2} \)
83 \( 1 + (-1.79 + 0.357i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (-1.39 - 0.576i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (2.96 - 2.96i)T - 97iT^{2} \)
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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

−12.08038268655804703663256079344, −10.79031505981471412830108407674, −10.62174399191309781266141204553, −9.607287931029039765091274734136, −8.764773248328989473373384727401, −7.56488450678056989057337208901, −6.44772232925393103124057364472, −5.66742111166510587481636439486, −3.75024822572083271014703708319, −2.91863307510196558617190070295, 0.03652598227493566837712879231, 1.42965863817234175047589012879, 3.45343317357015022593672469220, 5.28084088746736332222850426627, 6.52196116728509275311770899300, 7.27189805411142550806563113273, 7.78248697896609192063969835113, 9.433067749554241690508110157721, 9.695263089041393433122836229083, 11.10494000104324628100487934329

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