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Properties

Label 1-1792-1792.499-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.156 - 0.987i$
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  + (−0.973 − 0.227i)3-s + (−0.812 + 0.582i)5-s + (0.896 + 0.442i)9-s + (−0.0327 + 0.999i)11-s + (−0.995 − 0.0980i)13-s + (0.923 − 0.382i)15-s + (0.793 − 0.608i)17-s + (0.935 + 0.352i)19-s + (−0.946 + 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.773 − 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.986 + 0.162i)37-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)3-s + (−0.812 + 0.582i)5-s + (0.896 + 0.442i)9-s + (−0.0327 + 0.999i)11-s + (−0.995 − 0.0980i)13-s + (0.923 − 0.382i)15-s + (0.793 − 0.608i)17-s + (0.935 + 0.352i)19-s + (−0.946 + 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.773 − 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.986 + 0.162i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.156 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07764824719 - 0.09095478057i\)
\(L(\frac12)\) \(\approx\) \(0.07764824719 - 0.09095478057i\)
\(L(1)\) \(\approx\) \(0.5786851722 + 0.09751449615i\)
\(L(1)\) \(\approx\) \(0.5786851722 + 0.09751449615i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.973 - 0.227i)T \)
5 \( 1 + (-0.812 + 0.582i)T \)
11 \( 1 + (-0.0327 + 0.999i)T \)
13 \( 1 + (-0.995 - 0.0980i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (0.935 + 0.352i)T \)
23 \( 1 + (-0.946 + 0.321i)T \)
29 \( 1 + (-0.471 + 0.881i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.986 + 0.162i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (-0.290 + 0.956i)T \)
47 \( 1 + (-0.991 + 0.130i)T \)
53 \( 1 + (0.999 + 0.0327i)T \)
59 \( 1 + (-0.412 + 0.910i)T \)
61 \( 1 + (-0.729 - 0.683i)T \)
67 \( 1 + (-0.227 + 0.973i)T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (0.997 - 0.0654i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (0.634 + 0.773i)T \)
89 \( 1 + (-0.751 + 0.659i)T \)
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

−20.14221888264848555805414889131, −19.51008320183745726025714454651, −18.70629164918812500373172071930, −18.03285838685050911993444726539, −16.92881583677094429490945344480, −16.68953483389978122085687617827, −15.943846714304931009655167031479, −15.27197690973658302140194094420, −14.398584883706307257267800865542, −13.37099963118373788761542161182, −12.55123916401498806033698708697, −11.84453391448238827540901174260, −11.50516249058103750025246157077, −10.47847336241933216859423240097, −9.80873381042099658158661116081, −8.87950459217623460856786713068, −7.939105291817571694544641950947, −7.31050096939385798758242379307, −6.260780824888639010944364065302, −5.45737054229185483663161596522, −4.826782307737666986841795783650, −3.92593159495785370572541529400, −3.17852169161573864626620989430, −1.64911996849210156834807859646, −0.612100030929670692817678910181, 0.04262780656018711268046378537, 1.23247319285176831940931887624, 2.32676303909339961335881049734, 3.39163160148846910937938221704, 4.37763664191998213837982868602, 5.08620532010614269477824136512, 5.928682511687262677955720594797, 6.98562365193229039623286410995, 7.45074474915620373258897235207, 7.99917647846934105618556699972, 9.69995850474663616998835185417, 9.91620331980805063028204900261, 10.948345309771578968908361791486, 11.78059462707284037808640433323, 12.08407906785988891968967124928, 12.858333916596112005411323939106, 13.922431135111320683689218281047, 14.82735915945306630110288200621, 15.3337909497707255419133622080, 16.38891964229235832151342515645, 16.66396445696108434895278256663, 17.89951267492623063340297935398, 18.174439409165979915591485429570, 18.923762419352838967173036172710, 19.84551335498382311380576990775

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