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Properties

Label 1-1792-1792.331-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: $\chi_{1792} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.996 - 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09895482383 + 2.249597788i\)
\(L(\frac12)\) \(\approx\) \(-0.09895482383 + 2.249597788i\)
\(L(1)\) \(\approx\) \(1.208495191 + 0.6988610024i\)
\(L(1)\) \(\approx\) \(1.208495191 + 0.6988610024i\)

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

−19.772863778105380599127913882169, −19.20669879633032843517125973346, −17.977654213210807025460558186246, −17.68985976359587036277852618230, −16.62552573615385006420659776423, −15.96930697988212348109537360631, −14.998609984734415157238590133861, −14.59021578183524793657210165795, −13.48191348642116639120330607712, −13.002277661988772625923311505865, −12.46601327585105207833541097783, −11.53561700124692487005482102279, −10.541886302119506286411758212406, −9.51330983860457510287814716236, −8.92829824697833802521135214012, −8.34743907112255676111447841964, −7.640956738174957915116065616495, −6.50986480771673480584207184670, −5.990739799716801840129751632088, −4.46957467610642398253033925890, −4.22451823970536457136191046830, −3.01109307918797423978184585437, −2.07128085169602747984211864432, −1.23087817813864808262034619239, −0.32289351136836766852776648276, 1.54387273746795187931726676266, 2.200276413221540546040350597604, 3.2446637716496249908117709428, 3.88945302673877150133439737535, 4.58061185498480134727983845178, 5.95165072233183670538590453940, 6.76125323374606776702672580324, 7.32148788313481584601107225190, 8.45842233115251084015682388625, 9.03795018132284255265427544184, 9.732910567076801314421983805866, 10.70975484996517142852245822474, 11.17397977067635132192919997992, 12.12417183970066093852885277375, 13.27895910710832760792170420236, 13.90246259502196800203427048669, 14.45127876220522277246978387285, 15.12566405179138348291895676926, 15.79864128218707317048211577203, 16.61901910468256730599343986412, 17.505691425232417599548541729245, 18.38817943327824060580123133845, 19.06834118277254187943153792697, 19.61217435973789465121661320110, 20.26751654134358153469114167814

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