Properties

Label 1-1792-1792.1003-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.0878 + 0.996i$
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  + (−0.935 + 0.352i)3-s + (−0.683 + 0.729i)5-s + (0.751 − 0.659i)9-s + (−0.812 − 0.582i)11-s + (0.956 + 0.290i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.528 + 0.849i)19-s + (0.997 + 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (0.0327 + 0.999i)37-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)3-s + (−0.683 + 0.729i)5-s + (0.751 − 0.659i)9-s + (−0.812 − 0.582i)11-s + (0.956 + 0.290i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.528 + 0.849i)19-s + (0.997 + 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (0.0327 + 0.999i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.0878 + 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.0878 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6790881744 + 0.7415802308i\)
\(L(\frac12)\) \(\approx\) \(0.6790881744 + 0.7415802308i\)
\(L(1)\) \(\approx\) \(0.6812237395 + 0.1846703041i\)
\(L(1)\) \(\approx\) \(0.6812237395 + 0.1846703041i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.935 + 0.352i)T \)
5 \( 1 + (-0.683 + 0.729i)T \)
11 \( 1 + (-0.812 - 0.582i)T \)
13 \( 1 + (0.956 + 0.290i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.528 + 0.849i)T \)
23 \( 1 + (0.997 + 0.0654i)T \)
29 \( 1 + (-0.995 + 0.0980i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (0.0327 + 0.999i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.773 + 0.634i)T \)
47 \( 1 + (0.793 + 0.608i)T \)
53 \( 1 + (0.582 - 0.812i)T \)
59 \( 1 + (0.227 - 0.973i)T \)
61 \( 1 + (0.986 + 0.162i)T \)
67 \( 1 + (-0.352 - 0.935i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (0.881 - 0.471i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
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

−19.83549111542965278913357160276, −18.85802437993871043369234045909, −18.4511945991220545252180600008, −17.58289473852506393173316670278, −16.78071697280179582190992065219, −16.332554135816482745145398710014, −15.45726107311276249351556959712, −14.96761818774486653663555293214, −13.45936583152971878934346108280, −13.01999173496184133895270690036, −12.4235505248785832717016533864, −11.58360014099780830220007894120, −10.95664541722594659573447076917, −10.245298236170430658823925895991, −9.16038384486549115818565629024, −8.33751803187232247522660076709, −7.45094229889316863380138912353, −6.98955573547282116217573819367, −5.65367815321886604186937861541, −5.30007856454504984213438820947, −4.37836756064901187913424897462, −3.51552329746678948605580329113, −2.185766553585948247487996275694, −1.11634237720280920083946079069, −0.36623240877700599513391469314, 0.63832747856628350781554497908, 1.783156910940520550207383268159, 3.36812885515461940800203041693, 3.58364853422043538730248740071, 4.77566344272442793818817572343, 5.626955151615080098039114669738, 6.30005455717973517206776880678, 7.14715380560882886889375877857, 7.95464334640548721635931754783, 8.81335648020188802284635146851, 9.97247837173358022286374691362, 10.61495744643834898642376450791, 11.15145107131712940912179253269, 11.78901151319515317866481417161, 12.66799793576863045073586672814, 13.40603378177299653466795925771, 14.54587656121438198199592390801, 15.09527893040566955658499338488, 15.938894264058694334795360536856, 16.448632812147543636777870132713, 17.12513237543197040988588763592, 18.248860981728200350949297148832, 18.68782711378717088129498955936, 19.0868381732433631917484627144, 20.46311979540430436498134842685

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