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Properties

Label 2-320-320.43-c1-0-4
Degree $2$
Conductor $320$
Sign $0.935 - 0.354i$
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  + (0.301 − 1.38i)2-s + (−1.60 − 1.07i)3-s + (−1.81 − 0.832i)4-s + (0.316 + 2.21i)5-s + (−1.96 + 1.89i)6-s + (1.75 + 4.22i)7-s + (−1.69 + 2.26i)8-s + (0.275 + 0.666i)9-s + (3.15 + 0.229i)10-s + (−0.898 + 4.51i)11-s + (2.02 + 3.28i)12-s + (−0.163 − 0.822i)13-s + (6.36 − 1.14i)14-s + (1.86 − 3.88i)15-s + (2.61 + 3.02i)16-s − 5.91i·17-s + ⋯
L(s)  = 1  + (0.213 − 0.977i)2-s + (−0.926 − 0.618i)3-s + (−0.909 − 0.416i)4-s + (0.141 + 0.989i)5-s + (−0.801 + 0.772i)6-s + (0.661 + 1.59i)7-s + (−0.600 + 0.799i)8-s + (0.0919 + 0.222i)9-s + (0.997 + 0.0724i)10-s + (−0.271 + 1.36i)11-s + (0.584 + 0.948i)12-s + (−0.0453 − 0.228i)13-s + (1.70 − 0.306i)14-s + (0.481 − 1.00i)15-s + (0.653 + 0.757i)16-s − 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.935 - 0.354i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.935 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808142 + 0.147896i\)
\(L(\frac12)\) \(\approx\) \(0.808142 + 0.147896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.301 + 1.38i)T \)
5 \( 1 + (-0.316 - 2.21i)T \)
good3 \( 1 + (1.60 + 1.07i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-1.75 - 4.22i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.898 - 4.51i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (0.163 + 0.822i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + 5.91iT - 17T^{2} \)
19 \( 1 + (2.49 - 3.72i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.62 + 0.671i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.38 - 6.96i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 - 5.51T + 31T^{2} \)
37 \( 1 + (-3.63 - 0.722i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.787 - 1.90i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-0.161 - 0.241i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 - 1.99T + 47T^{2} \)
53 \( 1 + (4.95 + 7.41i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (4.78 - 3.19i)T + (22.5 - 54.5i)T^{2} \)
61 \( 1 + (-1.37 - 6.88i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (-6.12 + 9.16i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (0.870 - 2.10i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.90 - 3.27i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (5.74 + 5.74i)T + 79iT^{2} \)
83 \( 1 + (2.93 + 14.7i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (0.912 + 0.378i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-8.70 - 8.70i)T + 97iT^{2} \)
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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

−11.90480040738229124793365059837, −11.03961025078297481720776126007, −10.08178909475317990585025076397, −9.168904845191797412835188413644, −7.87290577469878002155206672614, −6.59113776656240700871888349026, −5.62435946605850691046604995381, −4.79079509487253657280548711895, −2.86699720644546360802347070683, −1.88734934861669517524484355950, 0.63352237797268090975586971388, 4.06580817259419892664718998102, 4.53779536677460377171549846849, 5.62500328947430526197836689025, 6.42147203945387812002725657466, 7.937279175061033609060630370413, 8.392213488385285827368826537822, 9.755737679465393210038181464507, 10.68397221708753751077874150563, 11.42011925172724963067932187806

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