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Properties

Label 1-80e2-6400.77-r1-0-0
Degree $1$
Conductor $6400$
Sign $-0.968 + 0.249i$
Analytic cond. $687.775$
Root an. cond. $687.775$
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  + (−0.327 + 0.944i)3-s + (0.831 + 0.555i)7-s + (−0.785 − 0.619i)9-s + (0.984 − 0.175i)11-s + (−0.214 − 0.976i)13-s + (0.233 + 0.972i)17-s + (0.967 + 0.252i)19-s + (−0.797 + 0.603i)21-s + (−0.488 + 0.872i)23-s + (0.842 − 0.539i)27-s + (−0.436 − 0.899i)29-s + (−0.156 + 0.987i)31-s + (−0.156 + 0.987i)33-s + (−0.931 − 0.364i)37-s + (0.993 + 0.117i)39-s + ⋯
L(s)  = 1  + (−0.327 + 0.944i)3-s + (0.831 + 0.555i)7-s + (−0.785 − 0.619i)9-s + (0.984 − 0.175i)11-s + (−0.214 − 0.976i)13-s + (0.233 + 0.972i)17-s + (0.967 + 0.252i)19-s + (−0.797 + 0.603i)21-s + (−0.488 + 0.872i)23-s + (0.842 − 0.539i)27-s + (−0.436 − 0.899i)29-s + (−0.156 + 0.987i)31-s + (−0.156 + 0.987i)33-s + (−0.931 − 0.364i)37-s + (0.993 + 0.117i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(687.775\)
Root analytic conductor: \(687.775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6400,\ (1:\ ),\ -0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2433255108 + 1.918545478i\)
\(L(\frac12)\) \(\approx\) \(0.2433255108 + 1.918545478i\)
\(L(1)\) \(\approx\) \(0.9709758472 + 0.4723876824i\)
\(L(1)\) \(\approx\) \(0.9709758472 + 0.4723876824i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.327 + 0.944i)T \)
7 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (0.984 - 0.175i)T \)
13 \( 1 + (-0.214 - 0.976i)T \)
17 \( 1 + (0.233 + 0.972i)T \)
19 \( 1 + (0.967 + 0.252i)T \)
23 \( 1 + (-0.488 + 0.872i)T \)
29 \( 1 + (-0.436 - 0.899i)T \)
31 \( 1 + (-0.156 + 0.987i)T \)
37 \( 1 + (-0.931 - 0.364i)T \)
41 \( 1 + (0.117 + 0.993i)T \)
43 \( 1 + (-0.290 + 0.956i)T \)
47 \( 1 + (0.522 + 0.852i)T \)
53 \( 1 + (0.137 - 0.990i)T \)
59 \( 1 + (0.916 - 0.400i)T \)
61 \( 1 + (0.999 + 0.0196i)T \)
67 \( 1 + (-0.603 + 0.797i)T \)
71 \( 1 + (0.346 + 0.938i)T \)
73 \( 1 + (-0.271 - 0.962i)T \)
79 \( 1 + (-0.233 + 0.972i)T \)
83 \( 1 + (0.967 + 0.252i)T \)
89 \( 1 + (-0.993 - 0.117i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−17.08457603664778260438898016587, −16.73580798493132626440271566966, −16.074907556167595000088264652925, −14.96944622313305376756153969517, −14.291740669073153156740564739933, −13.89407527912572342724420887151, −13.437862652121316331267514774826, −12.25992541723439283265576712712, −11.96147980786156618419869545651, −11.42231878283484001143767816108, −10.74564031099769705581564745693, −9.88820047864819980101616399887, −9.002231690071449569473042888441, −8.53389296723468993771399372927, −7.44411456167392693340873274304, −7.20390592610750319507466677485, −6.619308915943531191684593755188, −5.63783534684920047389538323918, −5.02862079925327671084540162225, −4.26476483142827494773926015124, −3.47207927834788802183686451128, −2.331061712526490046579220806193, −1.78108600963755064369685777236, −1.003099978317214801539342627808, −0.30712280776454154453931749729, 0.94865203503568855723238781371, 1.641300135684686536115132417433, 2.7481047359155345580389649209, 3.54809814587670216252860333347, 4.05966456617837777999613209262, 4.99551325926683345496296739606, 5.56656107685243204807478339579, 5.999161947430935738218976298872, 6.989151155382021214602553413070, 8.05972818937429786540358659392, 8.34186055649856636471743717988, 9.31928844511272699197408915571, 9.75731316754010751909140949531, 10.51179425060573387115497346751, 11.21581146803070372956151908411, 11.754965117687834657196041581632, 12.24585841092313664418321728504, 13.13645825881268677889818372819, 14.15464945361612514616515169805, 14.583740890812936551225201529133, 15.105584874969181733256043263056, 15.80782783566355383338588356836, 16.326465328555686909365259112879, 17.24067954781898651793786992138, 17.60752138341233019694329876049

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