Properties

Label 1-1792-1792.1461-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.585 + 0.810i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.057956453 + 1.051840477i\)
\(L(\frac12)\) \(\approx\) \(2.057956453 + 1.051840477i\)
\(L(1)\) \(\approx\) \(1.301238643 + 0.05741731322i\)
\(L(1)\) \(\approx\) \(1.301238643 + 0.05741731322i\)

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

−19.63140231667836073281940310140, −19.16421244449982154767607519452, −18.61997533590870579373916691337, −17.95309248344370784083183284907, −17.08638741140471718139584321911, −16.01786404230876173570784729282, −15.30003882130164893334231840329, −14.650752560708404736318190285753, −14.039811174784723611654465291610, −13.38212143811862677497142322995, −12.554201677457667086099322831415, −11.71449115264989919961047471419, −10.846253072045975029037228935098, −9.995504025784742959042195142932, −9.40618023664917531645495115003, −8.35860654323607559191237462431, −7.64837667558758820019628903789, −7.075748652336556188175708209017, −6.27245705827336029682623909276, −5.249170864488892765054651104338, −4.02386553490147944708256857489, −3.336384251411453099649166941738, −2.422232151851925641853622822556, −1.90410582808267817420992504133, −0.41111671047298452555123302266, 0.83371774517585661915460716059, 1.936762316165115388070880317832, 2.82815005414844326995049690563, 3.64252564057726700106474013486, 4.81689487374834152726784591493, 5.02131808156071686608287264864, 6.22923614074193317531600292277, 7.524108654513367442096677008956, 8.139130389961895292321537924368, 8.56876575050755065559625363614, 9.62332302169912872080154605870, 10.174301504986302326952272832611, 10.87570857427728132382807332335, 12.25440197956747813901221150935, 12.7972286772164547490707547091, 13.275515103143447466961712279882, 14.36431875536051215395572759593, 14.940165681940948710930244719453, 15.65632170669816586638046786533, 16.47361419119477566915304910278, 16.89095654413436214855598276996, 18.0851141427317476522408942950, 18.69080646802101870727789621730, 19.74204278012659392136292619575, 20.06917241959836171453966226446

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