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Properties

Label 1-1792-1792.853-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.999 + 0.0245i$
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  + (−0.634 + 0.773i)3-s + (0.956 − 0.290i)5-s + (−0.195 − 0.980i)9-s + (−0.995 − 0.0980i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (0.881 − 0.471i)19-s + (−0.555 + 0.831i)23-s + (0.831 − 0.555i)25-s + (0.881 + 0.471i)27-s + (−0.0980 − 0.995i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.471 + 0.881i)37-s + ⋯
L(s)  = 1  + (−0.634 + 0.773i)3-s + (0.956 − 0.290i)5-s + (−0.195 − 0.980i)9-s + (−0.995 − 0.0980i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (0.881 − 0.471i)19-s + (−0.555 + 0.831i)23-s + (0.831 − 0.555i)25-s + (0.881 + 0.471i)27-s + (−0.0980 − 0.995i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.471 + 0.881i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 + 0.0245i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.886494010 + 0.02315192677i\)
\(L(\frac12)\) \(\approx\) \(1.886494010 + 0.02315192677i\)
\(L(1)\) \(\approx\) \(1.007965251 + 0.1139277351i\)
\(L(1)\) \(\approx\) \(1.007965251 + 0.1139277351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.634 + 0.773i)T \)
5 \( 1 + (0.956 - 0.290i)T \)
11 \( 1 + (-0.995 - 0.0980i)T \)
13 \( 1 + (0.290 - 0.956i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (0.881 - 0.471i)T \)
23 \( 1 + (-0.555 + 0.831i)T \)
29 \( 1 + (-0.0980 - 0.995i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.471 + 0.881i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.634 - 0.773i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (-0.0980 + 0.995i)T \)
59 \( 1 + (0.290 + 0.956i)T \)
61 \( 1 + (0.773 + 0.634i)T \)
67 \( 1 + (0.773 + 0.634i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (0.980 - 0.195i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.471 + 0.881i)T \)
89 \( 1 + (0.555 + 0.831i)T \)
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

−20.071347617811705027642098503672, −18.87815314786453246284976973311, −18.40306494204812963479468955255, −18.07451703926600336383668677060, −17.15600346970086620016218005047, −16.373000643835694605000516880338, −15.93356940548837718986470602434, −14.462048156619081653829234847363, −14.0594185363867986519852800674, −13.268767348838528001199170565463, −12.68688914536616820576872521547, −11.72605575439929688306778236732, −11.15690540823349669158651978424, −10.17138648609232723389894788051, −9.692116351718435905767343369, −8.51224949552877938296941004586, −7.69868906894276336627104064449, −6.81761757112159015960822193501, −6.314221529165601186193963639765, −5.29232133461951574057633866783, −4.92851974970619592359223912818, −3.37098873187615098795852581614, −2.363681009170074307161281307852, −1.73063667662706917608309973579, −0.66091449938103203094894461358, 0.54011814780385182190991277966, 1.526032164205015777385560828172, 2.80165406918971616379736605155, 3.55227488359729497133766625022, 4.72145731351026242473387355243, 5.47184112519378039050986609457, 5.80221777807514498213592645290, 6.80037568358054445586117096300, 8.01168838437698319344536069040, 8.69880025869961790907599239538, 9.82809575500621103987295677044, 10.11331807315224934262080518260, 10.8056626195843053074291022698, 11.77689294236181473876961136834, 12.5408555594313673692904171573, 13.40028055949647879568226010639, 13.93178564548860691200311108468, 15.21947332129691732352546483381, 15.502085334188664878064516036092, 16.39170454477023290638601646214, 17.13818792281733239881938831944, 17.76704469023728060261975314608, 18.1896721959919944243947520594, 19.30274500324863563417918391784, 20.414359557062437944332835906691

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