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Properties

Label 1-40e2-1600.179-r1-0-0
Degree $1$
Conductor $1600$
Sign $-0.709 - 0.704i$
Analytic cond. $171.943$
Root an. cond. $171.943$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.996 + 0.0784i)3-s + (−0.707 + 0.707i)7-s + (0.987 + 0.156i)9-s + (0.233 − 0.972i)11-s + (0.972 − 0.233i)13-s + (−0.951 − 0.309i)17-s + (0.649 − 0.760i)19-s + (−0.760 + 0.649i)21-s + (−0.156 − 0.987i)23-s + (0.972 + 0.233i)27-s + (−0.996 − 0.0784i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.522 + 0.852i)37-s + (0.987 − 0.156i)39-s + ⋯
L(s)  = 1  + (0.996 + 0.0784i)3-s + (−0.707 + 0.707i)7-s + (0.987 + 0.156i)9-s + (0.233 − 0.972i)11-s + (0.972 − 0.233i)13-s + (−0.951 − 0.309i)17-s + (0.649 − 0.760i)19-s + (−0.760 + 0.649i)21-s + (−0.156 − 0.987i)23-s + (0.972 + 0.233i)27-s + (−0.996 − 0.0784i)29-s + (0.309 − 0.951i)31-s + (0.309 − 0.951i)33-s + (−0.522 + 0.852i)37-s + (0.987 − 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(171.943\)
Root analytic conductor: \(171.943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (1:\ ),\ -0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5237606472 - 1.271525879i\)
\(L(\frac12)\) \(\approx\) \(0.5237606472 - 1.271525879i\)
\(L(1)\) \(\approx\) \(1.263619157 - 0.1235722580i\)
\(L(1)\) \(\approx\) \(1.263619157 - 0.1235722580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.996 + 0.0784i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.233 - 0.972i)T \)
13 \( 1 + (0.972 - 0.233i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.649 - 0.760i)T \)
23 \( 1 + (-0.156 - 0.987i)T \)
29 \( 1 + (-0.996 - 0.0784i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.522 + 0.852i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.760 + 0.649i)T \)
59 \( 1 + (0.522 - 0.852i)T \)
61 \( 1 + (-0.852 + 0.522i)T \)
67 \( 1 + (-0.760 - 0.649i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + (-0.156 - 0.987i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.649 + 0.760i)T \)
89 \( 1 + (-0.987 + 0.156i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.457493493844600495467187247160, −19.84560067927928810586667665257, −19.30380111618048636806508162594, −18.38786322803427151942216974779, −17.73180266159719550396666351038, −16.78009959978138902106057154775, −15.845597501263217326428283171149, −15.45850089433306500288014350393, −14.43037750206254197866967065128, −13.80233031378050018042181979450, −13.14685693449859958914202084781, −12.52538101304418596717601695943, −11.47834364472751277933106605739, −10.43361129711030754774154706651, −9.81340344671526068593572422805, −9.09273040442781445345150357191, −8.33326231798017319053375017583, −7.27825114257213649556804380866, −6.92982184692929654104676627552, −5.86911101450205557914106546463, −4.57712003149942705364486231688, −3.74650963447648723891147853261, −3.30046919600527388090312231568, −1.936099191978379316945357620572, −1.36127897055734221742260317012, 0.19955730742582423680674294328, 1.42947045771202977800731740342, 2.58694015926854653172304980330, 3.143865471312053383941828820571, 3.97328996945844739774616485932, 5.014073640334698515295182877905, 6.15802410117250371669265169100, 6.68984859366565074163927809157, 7.8633736713567853605053262188, 8.61864498674441143203797443384, 9.123444988781810773135975982, 9.83955047784645188441408331366, 10.907281645408565348091068544597, 11.59445588449227987610002668039, 12.71318068570935764254139919141, 13.37996009236476580654844197655, 13.81131647246498308899834025015, 14.851030962609975339752493849098, 15.58247121761125647016739647888, 16.00356384777597396693105585271, 16.85399749069264018565992956344, 18.142955772489791690072217965613, 18.602272593228239794650342228274, 19.25235107157109043671999854116, 20.03553722732178037965888678581

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