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Properties

Label 1-40e2-1600.1389-r0-0-0
Degree $1$
Conductor $1600$
Sign $0.0274 + 0.999i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (−0.852 + 0.522i)3-s + (0.707 − 0.707i)7-s + (0.453 − 0.891i)9-s + (0.996 − 0.0784i)11-s + (−0.0784 + 0.996i)13-s + (0.587 + 0.809i)17-s + (−0.972 + 0.233i)19-s + (−0.233 + 0.972i)21-s + (−0.891 + 0.453i)23-s + (0.0784 + 0.996i)27-s + (−0.852 + 0.522i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.649 − 0.760i)37-s + (−0.453 − 0.891i)39-s + ⋯
L(s)  = 1  + (−0.852 + 0.522i)3-s + (0.707 − 0.707i)7-s + (0.453 − 0.891i)9-s + (0.996 − 0.0784i)11-s + (−0.0784 + 0.996i)13-s + (0.587 + 0.809i)17-s + (−0.972 + 0.233i)19-s + (−0.233 + 0.972i)21-s + (−0.891 + 0.453i)23-s + (0.0784 + 0.996i)27-s + (−0.852 + 0.522i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)33-s + (−0.649 − 0.760i)37-s + (−0.453 − 0.891i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0274 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0274 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.0274 + 0.999i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ 0.0274 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7546934059 + 0.7757295802i\)
\(L(\frac12)\) \(\approx\) \(0.7546934059 + 0.7757295802i\)
\(L(1)\) \(\approx\) \(0.8541679457 + 0.2177062997i\)
\(L(1)\) \(\approx\) \(0.8541679457 + 0.2177062997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.852 + 0.522i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (0.996 - 0.0784i)T \)
13 \( 1 + (-0.0784 + 0.996i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.972 + 0.233i)T \)
23 \( 1 + (-0.891 + 0.453i)T \)
29 \( 1 + (-0.852 + 0.522i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.649 - 0.760i)T \)
41 \( 1 + (-0.453 + 0.891i)T \)
43 \( 1 + (0.382 + 0.923i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.233 + 0.972i)T \)
59 \( 1 + (-0.649 - 0.760i)T \)
61 \( 1 + (0.760 + 0.649i)T \)
67 \( 1 + (0.233 + 0.972i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (0.891 - 0.453i)T \)
79 \( 1 + (-0.587 + 0.809i)T \)
83 \( 1 + (0.972 - 0.233i)T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−20.30521420156507411700945460245, −19.34179168826883981335012305817, −18.695338193908142240354536617921, −18.00877363174579436199395284744, −17.32887048339391504031050987375, −16.83054491394422572570190370802, −15.78554035043980418521816937800, −15.123057545683303713952488667654, −14.22864849466237309081043813253, −13.48829243061164290669200772451, −12.37501869175381389344361556583, −12.097460364839523100586873878431, −11.32126364951122386693123421514, −10.52015535070090303429212055714, −9.69494334162820438258540009572, −8.55476085210893390299169336343, −7.9783325109960208334994354320, −6.9802483191446457450864026422, −6.25655858304154941645472895403, −5.40394184619013679529071869918, −4.827622557312658358250778158097, −3.7154360201545214854482526396, −2.401376268735042755640418610177, −1.65458411433691733685072337563, −0.49294543565883100030498164587, 1.16024458160354455580915394053, 1.86835682089233105914372542171, 3.64258282384642743769740718857, 4.1245606721061761177545540212, 4.82773852245170898879823041934, 5.96762529707856139651058254413, 6.492512637524328764692035141087, 7.44749425475380737298101599462, 8.39178980486519824991688267741, 9.37539650676147621520300437675, 10.0431735545256139732476871314, 10.921822978488235755368355532898, 11.45094502776334237496913242704, 12.15871951954162617795512234454, 13.00339297509017038404493308086, 14.238644488536763041061651917718, 14.51938495272730193516059995172, 15.47680284354767433748803835613, 16.47063399574029957734719461752, 16.96235987872173280892011807339, 17.39752330215950536216175726830, 18.308106023604389296158237860221, 19.20413770984360151989793520826, 19.91964607968552105942888842638, 20.958620633075177412761133213

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