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Properties

Label 2-320-320.187-c1-0-18
Degree $2$
Conductor $320$
Sign $0.314 + 0.949i$
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  + (−0.387 − 1.36i)2-s + (0.229 + 1.15i)3-s + (−1.69 + 1.05i)4-s + (−2.23 − 0.0761i)5-s + (1.48 − 0.759i)6-s + (−0.382 − 0.922i)7-s + (2.09 + 1.90i)8-s + (1.49 − 0.618i)9-s + (0.762 + 3.06i)10-s + (3.82 − 2.55i)11-s + (−1.60 − 1.71i)12-s + (2.98 − 4.46i)13-s + (−1.10 + 0.877i)14-s + (−0.424 − 2.59i)15-s + (1.77 − 3.58i)16-s + 0.181·17-s + ⋯
L(s)  = 1  + (−0.274 − 0.961i)2-s + (0.132 + 0.666i)3-s + (−0.849 + 0.527i)4-s + (−0.999 − 0.0340i)5-s + (0.604 − 0.309i)6-s + (−0.144 − 0.348i)7-s + (0.739 + 0.672i)8-s + (0.497 − 0.206i)9-s + (0.241 + 0.970i)10-s + (1.15 − 0.769i)11-s + (−0.463 − 0.496i)12-s + (0.828 − 1.23i)13-s + (−0.295 + 0.234i)14-s + (−0.109 − 0.670i)15-s + (0.444 − 0.895i)16-s + 0.0440·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.314 + 0.949i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.314 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.822228 - 0.593844i\)
\(L(\frac12)\) \(\approx\) \(0.822228 - 0.593844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.387 + 1.36i)T \)
5 \( 1 + (2.23 + 0.0761i)T \)
good3 \( 1 + (-0.229 - 1.15i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.382 + 0.922i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.82 + 2.55i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (-2.98 + 4.46i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 - 0.181T + 17T^{2} \)
19 \( 1 + (-0.270 - 1.35i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.815 - 0.337i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (5.01 + 3.35i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 - 11.0T + 31T^{2} \)
37 \( 1 + (9.06 - 6.05i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (3.56 - 1.47i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.94 + 1.18i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + 9.72iT - 47T^{2} \)
53 \( 1 + (11.7 + 2.34i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (0.119 + 0.0238i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-5.48 - 3.66i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-10.0 + 1.99i)T + (61.8 - 25.6i)T^{2} \)
71 \( 1 + (-6.17 - 2.55i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-2.72 - 1.12i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.66 - 4.66i)T + 79iT^{2} \)
83 \( 1 + (-3.54 + 5.29i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (0.364 - 0.879i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (-0.818 + 0.818i)T - 97iT^{2} \)
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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

−11.41754786463521823523976660323, −10.49477729994558463492223001322, −9.833235049664807356943074395159, −8.659333023830383414089205496633, −8.113491147272662497512978483303, −6.71003790226954493490174599235, −4.98924763385997247922583800013, −3.69658981524022635866728791174, −3.48034122056243243574137874897, −0.975434855355405938788174658295, 1.48394744039702289034488979159, 3.87842130367180518829083903680, 4.77012902124301493526404570331, 6.53534314760442466419809362789, 6.88277157503217380469316882438, 7.889310735763346826472572453529, 8.807476680054568411565828189104, 9.569587169474303501272273588938, 10.92365781806976326294758978269, 12.00500844903771742348408354761

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