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Properties

Label 1-1792-1792.1021-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: $\chi_{1792} (1021, \cdot )$
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

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

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