Properties

Label 1-1792-1792.1059-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0388 - 0.999i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.582 − 0.812i)3-s + (0.528 − 0.849i)5-s + (−0.321 − 0.946i)9-s + (−0.935 + 0.352i)11-s + (0.881 + 0.471i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (0.729 + 0.683i)19-s + (−0.896 − 0.442i)23-s + (−0.442 − 0.896i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯
L(s)  = 1  + (0.582 − 0.812i)3-s + (0.528 − 0.849i)5-s + (−0.321 − 0.946i)9-s + (−0.935 + 0.352i)11-s + (0.881 + 0.471i)13-s + (−0.382 − 0.923i)15-s + (0.608 + 0.793i)17-s + (0.729 + 0.683i)19-s + (−0.896 − 0.442i)23-s + (−0.442 − 0.896i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0388 - 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0388 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.340387872 - 2.251227002i\)
\(L(\frac12)\) \(\approx\) \(2.340387872 - 2.251227002i\)
\(L(1)\) \(\approx\) \(1.361893188 - 0.5851055279i\)
\(L(1)\) \(\approx\) \(1.361893188 - 0.5851055279i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.582 + 0.812i)T \)
5 \( 1 + (-0.528 + 0.849i)T \)
11 \( 1 + (0.935 - 0.352i)T \)
13 \( 1 + (-0.881 - 0.471i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (-0.729 - 0.683i)T \)
23 \( 1 + (0.896 + 0.442i)T \)
29 \( 1 + (-0.634 - 0.773i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.973 + 0.227i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (-0.995 + 0.0980i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (-0.352 - 0.935i)T \)
59 \( 1 + (-0.0327 + 0.999i)T \)
61 \( 1 + (-0.910 + 0.412i)T \)
67 \( 1 + (0.812 + 0.582i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.751 + 0.659i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.290 - 0.956i)T \)
89 \( 1 + (0.0654 - 0.997i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−20.39004189932468241319882678872, −19.49612994143259357667092943904, −18.6642823607910837961482699771, −18.11162116211183331044944044058, −17.34812601838737792437198366760, −16.129878831952605024863096977987, −15.872503300210092389872293569124, −15.026984036667789311784052930746, −14.29843306978100447087626971197, −13.48083418034364922611174534690, −13.27982557549084438643244245824, −11.61807805140946539076085047765, −11.16020449471548666494209532197, −10.14177426851299864580485393560, −9.90241547238889216718087616138, −8.954177212320202303012601727240, −7.93611173909841951309765930422, −7.51013851696397552309121826384, −6.14456863445934658331598400544, −5.595176493160475860044619007815, −4.65534701158791580868577349262, −3.58208556623121334930756835360, −2.87913945578002402290451169023, −2.332246546877038877878938841228, −0.84872947872323482813512118717, 0.646140498984737418328426667179, 1.516058694579467565222432610835, 2.16528666185717047738507626826, 3.27101590286256116453634477495, 4.15801525873388909731052725097, 5.3092144226813212667231455468, 5.97361430065513128762432546780, 6.80823168813646208253328495118, 7.89561576585391542771988606058, 8.314039955995754255005898197469, 9.09572488760920609479067816725, 9.944494396876247955350096788402, 10.702366186013826090501438264460, 12.0527648281095525534519738979, 12.40870886477765301223193668767, 13.14207073765305575758492952179, 13.91169434535850232040335696417, 14.31752070195281217842271800548, 15.47038173201845673269463696648, 16.18433441357615095558568422696, 16.910318433806262656523468368223, 17.991061666404380683418049984250, 18.18157978537555702700297846189, 19.08893306798462201045941019202, 19.94440822142378702740643727930

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