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Properties

Label 1-1792-1792.731-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.996 - 0.0857i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (−0.528 + 0.849i)3-s + (0.162 + 0.986i)5-s + (−0.442 − 0.896i)9-s + (−0.683 + 0.729i)11-s + (0.634 − 0.773i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.412 + 0.910i)19-s + (−0.321 + 0.946i)23-s + (−0.946 + 0.321i)25-s + (0.995 + 0.0980i)27-s + (0.956 + 0.290i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.812 + 0.582i)37-s + ⋯
L(s)  = 1  + (−0.528 + 0.849i)3-s + (0.162 + 0.986i)5-s + (−0.442 − 0.896i)9-s + (−0.683 + 0.729i)11-s + (0.634 − 0.773i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.412 + 0.910i)19-s + (−0.321 + 0.946i)23-s + (−0.946 + 0.321i)25-s + (0.995 + 0.0980i)27-s + (0.956 + 0.290i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.812 + 0.582i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.996 - 0.0857i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.996 - 0.0857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03873155434 + 0.9018138596i\)
\(L(\frac12)\) \(\approx\) \(-0.03873155434 + 0.9018138596i\)
\(L(1)\) \(\approx\) \(0.6914065483 + 0.4783886829i\)
\(L(1)\) \(\approx\) \(0.6914065483 + 0.4783886829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.528 + 0.849i)T \)
5 \( 1 + (0.162 + 0.986i)T \)
11 \( 1 + (-0.683 + 0.729i)T \)
13 \( 1 + (0.634 - 0.773i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (-0.412 + 0.910i)T \)
23 \( 1 + (-0.321 + 0.946i)T \)
29 \( 1 + (0.956 + 0.290i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (0.812 + 0.582i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (-0.471 + 0.881i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (-0.729 - 0.683i)T \)
59 \( 1 + (-0.352 + 0.935i)T \)
61 \( 1 + (-0.0327 + 0.999i)T \)
67 \( 1 + (0.849 + 0.528i)T \)
71 \( 1 + (-0.555 - 0.831i)T \)
73 \( 1 + (-0.0654 + 0.997i)T \)
79 \( 1 + (0.608 + 0.793i)T \)
83 \( 1 + (0.0980 + 0.995i)T \)
89 \( 1 + (-0.659 + 0.751i)T \)
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

−19.76922273766015802675357670552, −18.95938400802350256232009307056, −18.38981528599579021713385283707, −17.64765417494871703542450304296, −16.856447540999754144762023716372, −16.23115382466768573009454605927, −15.80029940819980433341733928896, −14.30369951896141503575291392793, −13.74334484831438709088483502727, −13.076032968061576487892389724277, −12.42698880713513822878143506760, −11.68119767493718107340584707849, −11.00934194252906706271302995927, −10.10032768079401646071144698230, −9.01874222329875584217040279264, −8.35701690982250886884882987776, −7.7425352401345053230552585341, −6.57503695139381212995854683592, −6.096690567594461515040818768158, −5.05613464308642147204041851922, −4.613910034400541862352315825212, −3.18060126476387559226992493424, −2.19668191834480232595085610757, −1.21929781201898156464104886805, −0.37441759925199535471398571593, 1.37078290296317948655600187061, 2.63950296155042080510277146528, 3.44591191588213969879001603951, 4.14801354267753990254160559046, 5.28002775552791123721479308161, 5.903369428952255809203526646093, 6.58877736920226131108513805703, 7.7279097529524735579614333842, 8.32884988998536056920311602058, 9.758579819607820403836864063747, 10.03643367627166423739484904611, 10.6996396746462371891959274500, 11.436517618886568148777211646427, 12.26863840862452976226983407960, 13.115171640583079396757498084123, 14.08911459831823542735722968032, 14.971013159250298427480831548966, 15.25298695565655981588364244543, 16.09945619009614462399632002942, 16.9097918759910577738675261010, 17.783645625829397952389844977530, 18.12327638399310644409809616149, 19.02463060330921823763402332476, 19.93037075528213460434299125140, 20.88706569135898765535224824000

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