Properties

Label 1-1792-1792.965-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.219 + 0.975i$
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  + (0.290 − 0.956i)3-s + (0.0980 − 0.995i)5-s + (−0.831 − 0.555i)9-s + (−0.881 − 0.471i)11-s + (0.995 − 0.0980i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (−0.773 − 0.634i)19-s + (−0.195 − 0.980i)23-s + (−0.980 − 0.195i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.634 − 0.773i)37-s + ⋯
L(s)  = 1  + (0.290 − 0.956i)3-s + (0.0980 − 0.995i)5-s + (−0.831 − 0.555i)9-s + (−0.881 − 0.471i)11-s + (0.995 − 0.0980i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (−0.773 − 0.634i)19-s + (−0.195 − 0.980i)23-s + (−0.980 − 0.195i)25-s + (−0.773 + 0.634i)27-s + (−0.471 − 0.881i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.634 − 0.773i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.219 + 0.975i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.219 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8014660820 - 1.001400186i\)
\(L(\frac12)\) \(\approx\) \(-0.8014660820 - 1.001400186i\)
\(L(1)\) \(\approx\) \(0.7482628356 - 0.7069232723i\)
\(L(1)\) \(\approx\) \(0.7482628356 - 0.7069232723i\)

Euler product

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

−20.79451110776231198738680411551, −19.85423718040884158958453702612, −19.04255961923404452065547508507, −18.40835529883718253552837018508, −17.648669450695758498447741834270, −16.74828016095791149908317659828, −15.97868981183076476816358590565, −15.33968378923517777975931891423, −14.7134267233711388880551754063, −14.07044306758325266560144496760, −13.299223724309680405082095852544, −12.30516751638919363318570284101, −11.26764301327239610705035584812, −10.60598005757663414872003769064, −10.21060673015198222985936094689, −9.35815282962443441215177554822, −8.399831617443013574251130441400, −7.72708299621034660687479688925, −6.78128896820201053693148118591, −5.72222247991530877558315356566, −5.23146537369446318958832617511, −3.83179934742747899224433503869, −3.55596933183576444061783002757, −2.52533632666843637149694633741, −1.60805104659363182820142336338, 0.25216611571522386048626473039, 0.84083660345559927090060737944, 1.903683346443818604417993277147, 2.72944421199125584171434761882, 3.765842010270948772379600593329, 4.78734326287204471124591360001, 5.85591679903262758650395711357, 6.16327689581234128055226426467, 7.58632205940986977330931224575, 7.93460462147234341115966662129, 8.8331430241384401464238294354, 9.32386156624567452866364932203, 10.60711989722889016597492077873, 11.308019397249660349196781075444, 12.30528554604050033279268507382, 12.82442875563483919733885805249, 13.4288737820687025714143942046, 14.0513273446973108978364431930, 15.0068753784524261218192159574, 15.9626340992691113223365650034, 16.53854675924970854055438555888, 17.38771219794236493294413637274, 18.10031056356916898837301525750, 18.8101257937572612784340442904, 19.369018191888750948812311817179

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