Properties

Label 1-1792-1792.1227-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.996 + 0.0878i$
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  + (−0.910 − 0.412i)3-s + (−0.0327 − 0.999i)5-s + (0.659 + 0.751i)9-s + (−0.986 + 0.162i)11-s + (−0.471 − 0.881i)13-s + (−0.382 + 0.923i)15-s + (−0.991 − 0.130i)17-s + (0.973 − 0.227i)19-s + (−0.0654 + 0.997i)23-s + (−0.997 + 0.0654i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.729 − 0.683i)37-s + ⋯
L(s)  = 1  + (−0.910 − 0.412i)3-s + (−0.0327 − 0.999i)5-s + (0.659 + 0.751i)9-s + (−0.986 + 0.162i)11-s + (−0.471 − 0.881i)13-s + (−0.382 + 0.923i)15-s + (−0.991 − 0.130i)17-s + (0.973 − 0.227i)19-s + (−0.0654 + 0.997i)23-s + (−0.997 + 0.0654i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.729 − 0.683i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.996 + 0.0878i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.996 + 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3453330099 + 0.01519043419i\)
\(L(\frac12)\) \(\approx\) \(0.3453330099 + 0.01519043419i\)
\(L(1)\) \(\approx\) \(0.5535294878 - 0.2102637890i\)
\(L(1)\) \(\approx\) \(0.5535294878 - 0.2102637890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.910 - 0.412i)T \)
5 \( 1 + (-0.0327 - 0.999i)T \)
11 \( 1 + (-0.986 + 0.162i)T \)
13 \( 1 + (-0.471 - 0.881i)T \)
17 \( 1 + (-0.991 - 0.130i)T \)
19 \( 1 + (0.973 - 0.227i)T \)
23 \( 1 + (-0.0654 + 0.997i)T \)
29 \( 1 + (-0.773 - 0.634i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (-0.729 - 0.683i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (-0.0980 + 0.995i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (-0.162 - 0.986i)T \)
59 \( 1 + (0.528 + 0.849i)T \)
61 \( 1 + (0.582 + 0.812i)T \)
67 \( 1 + (0.412 - 0.910i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (-0.946 - 0.321i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (-0.956 - 0.290i)T \)
89 \( 1 + (-0.896 - 0.442i)T \)
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

−20.15528729311824727506142255142, −18.919004958087359671385334291478, −18.521670040866507402706396347927, −17.86940176819312051673340329960, −17.10505596819187588023723039750, −16.20756350312984398178034206638, −15.75518471962180478087493796418, −14.86306945041217956954127669857, −14.25632414553531634742353045411, −13.2411272635367172689978923079, −12.470223315697485742985826423388, −11.47506547364735881405201374208, −11.10919453099930754484380685002, −10.30792237848818604222039448540, −9.708794366802768559579237021563, −8.74979689599598855369957488805, −7.51553806501345378479671855937, −6.942632628350873282940592620544, −6.17809425104221785719721519019, −5.33869242214962569756572926970, −4.53482485681458264005595187857, −3.63305531886948895050704866468, −2.68250188206582256798961916067, −1.67761155376286964799645270795, −0.14465794672419063461875884047, 0.44082384838785383797332513072, 1.5352631011276090593826702488, 2.41651802603397257717468542337, 3.73816141139007343383601998448, 4.847223406831126508884892909892, 5.32278710050140292424917402508, 5.90303150803498017401032019504, 7.277998565757026979651194970421, 7.57969245573460459582047779842, 8.58805163080263237317239517619, 9.56504119870057137811576668840, 10.26506927407635875073304481199, 11.22177674432477441757425094067, 11.77440507413360289926133200047, 12.78671518246416194348585267503, 13.02654277702756383844655383231, 13.762245842660796578770385886273, 15.11053857605241038218121953440, 15.87088274278184584654725119722, 16.24912552800523810089353890282, 17.27438892060973398457395595197, 17.77808355140455453649729963112, 18.25260903257100835121330612456, 19.43764208679758520370832248962, 19.920795095353541375904820498825

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