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Properties

Label 2-320-320.107-c1-0-12
Degree $2$
Conductor $320$
Sign $-0.231 - 0.972i$
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  + (−1.12 − 0.850i)2-s + (−1.38 + 2.07i)3-s + (0.553 + 1.92i)4-s + (−0.180 + 2.22i)5-s + (3.33 − 1.16i)6-s + (4.18 − 1.73i)7-s + (1.00 − 2.64i)8-s + (−1.24 − 3.00i)9-s + (2.09 − 2.36i)10-s + (−0.915 + 4.60i)11-s + (−4.76 − 1.51i)12-s + (3.52 − 0.701i)13-s + (−6.20 − 1.60i)14-s + (−4.38 − 3.47i)15-s + (−3.38 + 2.12i)16-s + 0.591·17-s + ⋯
L(s)  = 1  + (−0.798 − 0.601i)2-s + (−0.802 + 1.20i)3-s + (0.276 + 0.960i)4-s + (−0.0808 + 0.996i)5-s + (1.36 − 0.476i)6-s + (1.58 − 0.655i)7-s + (0.356 − 0.934i)8-s + (−0.414 − 1.00i)9-s + (0.663 − 0.747i)10-s + (−0.275 + 1.38i)11-s + (−1.37 − 0.438i)12-s + (0.977 − 0.194i)13-s + (−1.65 − 0.428i)14-s + (−1.13 − 0.896i)15-s + (−0.846 + 0.531i)16-s + 0.143·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.231 - 0.972i$
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: \((2,\ 320,\ (\ :1/2),\ -0.231 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453935 + 0.574900i\)
\(L(\frac12)\) \(\approx\) \(0.453935 + 0.574900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.850i)T \)
5 \( 1 + (0.180 - 2.22i)T \)
good3 \( 1 + (1.38 - 2.07i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (-4.18 + 1.73i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.915 - 4.60i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (-3.52 + 0.701i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 - 0.591T + 17T^{2} \)
19 \( 1 + (3.16 - 4.74i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.49 - 3.61i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.31 + 6.63i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + 0.228T + 31T^{2} \)
37 \( 1 + (-1.26 + 6.35i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.59 - 3.85i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.766 + 0.512i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 - 5.89iT - 47T^{2} \)
53 \( 1 + (3.44 - 2.30i)T + (20.2 - 48.9i)T^{2} \)
59 \( 1 + (-0.450 + 0.301i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-0.969 - 4.87i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (8.32 + 5.56i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (0.323 - 0.780i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.915 + 2.21i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-9.21 - 9.21i)T + 79iT^{2} \)
83 \( 1 + (-6.62 + 1.31i)T + (76.6 - 31.7i)T^{2} \)
89 \( 1 + (-4.86 - 2.01i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (6.30 - 6.30i)T - 97iT^{2} \)
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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

−11.32974309887941824898011183819, −10.89858714875674375873850187408, −10.28418448148802988010558475889, −9.580940553614155408768490889817, −8.044310016678354859857548694177, −7.46643486831034112561157826556, −5.97977182746180646966427929161, −4.47301694147454497077363156820, −3.83534802864427305203961952405, −1.89807987161753402453926600264, 0.803387433752627122705271285456, 1.86502775248795409797640409565, 4.87333055176660334744881507603, 5.64962999975549386421443168536, 6.45660554855487426256861963778, 7.73053513677305620026677998879, 8.523331730941180549130550207080, 8.842603793179543889500712964392, 10.81023943002314820852580816997, 11.30469476736169732626943285032

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