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Properties

Label 1-1792-1792.461-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.219 - 0.975i$
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  + (0.956 − 0.290i)3-s + (−0.995 + 0.0980i)5-s + (0.831 − 0.555i)9-s + (−0.471 − 0.881i)11-s + (0.0980 − 0.995i)13-s + (−0.923 + 0.382i)15-s + (0.923 + 0.382i)17-s + (0.634 + 0.773i)19-s + (0.195 − 0.980i)23-s + (0.980 − 0.195i)25-s + (0.634 − 0.773i)27-s + (0.881 + 0.471i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.773 − 0.634i)37-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)3-s + (−0.995 + 0.0980i)5-s + (0.831 − 0.555i)9-s + (−0.471 − 0.881i)11-s + (0.0980 − 0.995i)13-s + (−0.923 + 0.382i)15-s + (0.923 + 0.382i)17-s + (0.634 + 0.773i)19-s + (0.195 − 0.980i)23-s + (0.980 − 0.195i)25-s + (0.634 − 0.773i)27-s + (0.881 + 0.471i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.773 − 0.634i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.219 - 0.975i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.219 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.577855134 - 1.971467614i\)
\(L(\frac12)\) \(\approx\) \(1.577855134 - 1.971467614i\)
\(L(1)\) \(\approx\) \(1.262391595 - 0.3620232781i\)
\(L(1)\) \(\approx\) \(1.262391595 - 0.3620232781i\)

Euler product

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

−20.412879506107655375417172100874, −19.42745384081971716316149815714, −19.00583658085477201403341068602, −18.27425896632266715515512387005, −17.20760595235767192832052988384, −16.29502064514053315642474413759, −15.61992962390723281350893737547, −15.24625543503398258621762908678, −14.3027415740258950996481146602, −13.70587323084373732550331934938, −12.81211193227224630757632210065, −11.999384568910832419194688530511, −11.33463963087449648520053916460, −10.31284295767585424048567109759, −9.52146713749255285753537909218, −8.94877453971901963214828703569, −7.93516012092654144069534358766, −7.46082938059002818474901274539, −6.78479310097816237270471570038, −5.227240343130247103335862057089, −4.594680587267741355424420791914, −3.77232697364707876714489800125, −3.039537963521753599798563189578, −2.08707208039478731891281410422, −1.00026542397590599163685439296, 0.463300425742112748391306216145, 1.26946274310423175001538310868, 2.64837240918551017659364053592, 3.333812160264037073078075379601, 3.82680346263001260634651068221, 5.0577175481417096281953189538, 5.95318647058014039956709469259, 7.13826800717725011601256454093, 7.64463227892656030851689351775, 8.5119980775074384472578639257, 8.73694024099764208164448124919, 10.34295902283124995532446687466, 10.43115321009710185911505526845, 11.79019623118404467394878469972, 12.418681233589434758740099804218, 13.05421197603082497452665326169, 14.03268870825299460088977915066, 14.58866213518247896388206143421, 15.3088803932935455470076160837, 16.074655563925209719947240107521, 16.568021488363988042333743563190, 17.97737924564262879950569749796, 18.4481576263840394691354763676, 19.214623110148619030642256959329, 19.67477062347814649029785291349

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