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Properties

Label 1-1792-1792.963-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: $\chi_{1792} (963, \cdot )$
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

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

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