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Properties

Label 1-1792-1792.1011-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.416 - 0.909i$
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  + (−0.683 + 0.729i)3-s + (−0.910 − 0.412i)5-s + (−0.0654 − 0.997i)9-s + (−0.849 − 0.528i)11-s + (0.995 + 0.0980i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.162 + 0.986i)19-s + (0.751 + 0.659i)23-s + (0.659 + 0.751i)25-s + (0.773 + 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.352 − 0.935i)37-s + ⋯
L(s)  = 1  + (−0.683 + 0.729i)3-s + (−0.910 − 0.412i)5-s + (−0.0654 − 0.997i)9-s + (−0.849 − 0.528i)11-s + (0.995 + 0.0980i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.162 + 0.986i)19-s + (0.751 + 0.659i)23-s + (0.659 + 0.751i)25-s + (0.773 + 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.352 − 0.935i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5251061349 - 0.3370732374i\)
\(L(\frac12)\) \(\approx\) \(0.5251061349 - 0.3370732374i\)
\(L(1)\) \(\approx\) \(0.6665395571 + 0.02410517404i\)
\(L(1)\) \(\approx\) \(0.6665395571 + 0.02410517404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.683 + 0.729i)T \)
5 \( 1 + (-0.910 - 0.412i)T \)
11 \( 1 + (-0.849 - 0.528i)T \)
13 \( 1 + (0.995 + 0.0980i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (0.162 + 0.986i)T \)
23 \( 1 + (0.751 + 0.659i)T \)
29 \( 1 + (-0.471 + 0.881i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.352 - 0.935i)T \)
41 \( 1 + (0.980 + 0.195i)T \)
43 \( 1 + (-0.290 + 0.956i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (-0.528 + 0.849i)T \)
59 \( 1 + (0.582 + 0.812i)T \)
61 \( 1 + (0.227 - 0.973i)T \)
67 \( 1 + (-0.729 - 0.683i)T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (0.442 - 0.896i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (-0.634 - 0.773i)T \)
89 \( 1 + (-0.946 - 0.321i)T \)
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

−20.28364470183154539355849206602, −19.29497521643814646650254987208, −18.940057725764762667565456761998, −18.10564006055828545379223168916, −17.60943973200476999839848759108, −16.67101026555818164227910802580, −15.875188112907869884544601933782, −15.31636026464380058567420536260, −14.48261096205999134474835523012, −13.32228879074260305163165942795, −12.92605989108233744300155140928, −12.15665585362277187108013965298, −11.12842839175415545952786792186, −10.98625913213524461628258349114, −10.05315222984464173708280300587, −8.66697992253244656466871959354, −8.09037551894499010097499962347, −7.2446788829743451329178162955, −6.69630608584793166000084953227, −5.7869098064850960150438859643, −4.89561732938265472850014168601, −4.03667270706189393826849518001, −2.96788818752769020728061902894, −2.04556795290448842080391496453, −0.87035408369857392267244219747, 0.33305918536640840568718517712, 1.43736852755193422479513683143, 3.15866174902087762779253313940, 3.618221266553324336176340447385, 4.59662142121498592752198103895, 5.336337570178957424944918451595, 5.97297243552598658362681009775, 7.14410939945839878370941875234, 7.87189522167260327062905163155, 8.86626031901536502252213557512, 9.38131942658802029517345500605, 10.55259330072745140922825443829, 11.10905777341066474268653420793, 11.61766428019928246434647300813, 12.55567004814481580951184692724, 13.1997979528910035080827725967, 14.281078065543389524179540902549, 15.13977495341885449464551267695, 15.86815037791856865445624073273, 16.313144608364672011693683336567, 16.76709087382131675372874917692, 18.09160561525498958698232819059, 18.35379863599614242001000323468, 19.3619338009138461632778368940, 20.27235545767601923564816950153

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