Properties

Label 1-1792-1792.739-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: $\chi_{1792} (739, \cdot )$
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

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

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