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Properties

Label 1-1792-1792.349-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0245 + 0.999i$
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  + (0.995 + 0.0980i)3-s + (0.471 + 0.881i)5-s + (0.980 + 0.195i)9-s + (0.773 + 0.634i)11-s + (0.881 + 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (0.956 − 0.290i)19-s + (−0.831 + 0.555i)23-s + (−0.555 + 0.831i)25-s + (0.956 + 0.290i)27-s + (−0.634 − 0.773i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (0.290 − 0.956i)37-s + ⋯
L(s)  = 1  + (0.995 + 0.0980i)3-s + (0.471 + 0.881i)5-s + (0.980 + 0.195i)9-s + (0.773 + 0.634i)11-s + (0.881 + 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (0.956 − 0.290i)19-s + (−0.831 + 0.555i)23-s + (−0.555 + 0.831i)25-s + (0.956 + 0.290i)27-s + (−0.634 − 0.773i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (0.290 − 0.956i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0245 + 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0245 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.363666358 + 3.282106362i\)
\(L(\frac12)\) \(\approx\) \(3.363666358 + 3.282106362i\)
\(L(1)\) \(\approx\) \(1.808304688 + 0.6529779202i\)
\(L(1)\) \(\approx\) \(1.808304688 + 0.6529779202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.995 + 0.0980i)T \)
5 \( 1 + (0.471 + 0.881i)T \)
11 \( 1 + (0.773 + 0.634i)T \)
13 \( 1 + (0.881 + 0.471i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (0.956 - 0.290i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (-0.634 - 0.773i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.290 - 0.956i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (0.995 - 0.0980i)T \)
47 \( 1 + (0.923 + 0.382i)T \)
53 \( 1 + (-0.634 + 0.773i)T \)
59 \( 1 + (0.881 - 0.471i)T \)
61 \( 1 + (-0.0980 + 0.995i)T \)
67 \( 1 + (-0.0980 + 0.995i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.195 - 0.980i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.290 - 0.956i)T \)
89 \( 1 + (0.831 + 0.555i)T \)
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

−20.04722104986708466302631573310, −19.22202931599827555845109276302, −18.327107738006089798737692140871, −17.84434441935358972159251362307, −16.75573748001897008892732260630, −16.03037315442482124938615934665, −15.60510608701261549701521703773, −14.30678334360674741216672790250, −13.95467202515031160749067645416, −13.32200509261216045442417227224, −12.50245730065072742078735136449, −11.78519596031879428419248188835, −10.710064144603977353886720208478, −9.76153890805755400029442943160, −9.13832892373061494273992345404, −8.54997953503415154797524735985, −7.879729109455275151090191507853, −6.84032513376730939637369560752, −5.98238787902672528536823934891, −5.07812071055284572007887891972, −4.1001770928059131475773817198, −3.36448944730839711241517585994, −2.40209972820689123155462186966, −1.35177069229901466928139878039, −0.74727786456127651044334639195, 1.25173897862257246298269889377, 2.02428217448696988084956719875, 2.79520870214279049514452685191, 3.89983430464356659807135178439, 4.20101487653264607468626035974, 5.78264552349041743276743165219, 6.414836380183344857517918728882, 7.34527626151858941602867570696, 7.91399604715078419066857428677, 9.10420181757551185229496347815, 9.468293430273576556202678612664, 10.29313496364018831646114888508, 11.12433782864540729719999844864, 11.93672709346419300359964566681, 13.05720930385075647368853381614, 13.66261189871518837313357649735, 14.26419474196235427312082137845, 14.94522212865753303073823078126, 15.56315527247364425534901764313, 16.36931981672981217723351954177, 17.59212560601989733157017752400, 17.90413434097206555269858796330, 19.02057749620477991499468929994, 19.27601558957846051543962175432, 20.28192889933419861717553372451

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