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Properties

Label 1-1792-1792.1475-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.624 - 0.780i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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  + (−0.162 − 0.986i)3-s + (−0.973 + 0.227i)5-s + (−0.946 + 0.321i)9-s + (0.910 + 0.412i)11-s + (−0.956 + 0.290i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.999 + 0.0327i)19-s + (−0.442 + 0.896i)23-s + (0.896 − 0.442i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.849 + 0.528i)37-s + ⋯
L(s)  = 1  + (−0.162 − 0.986i)3-s + (−0.973 + 0.227i)5-s + (−0.946 + 0.321i)9-s + (0.910 + 0.412i)11-s + (−0.956 + 0.290i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.999 + 0.0327i)19-s + (−0.442 + 0.896i)23-s + (0.896 − 0.442i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.849 + 0.528i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2553022387 - 0.5312915474i\)
\(L(\frac12)\) \(\approx\) \(0.2553022387 - 0.5312915474i\)
\(L(1)\) \(\approx\) \(0.6971510928 - 0.1924578850i\)
\(L(1)\) \(\approx\) \(0.6971510928 - 0.1924578850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.162 - 0.986i)T \)
5 \( 1 + (-0.973 + 0.227i)T \)
11 \( 1 + (0.910 + 0.412i)T \)
13 \( 1 + (-0.956 + 0.290i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.999 + 0.0327i)T \)
23 \( 1 + (-0.442 + 0.896i)T \)
29 \( 1 + (-0.995 - 0.0980i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.849 + 0.528i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (-0.773 - 0.634i)T \)
47 \( 1 + (-0.130 - 0.991i)T \)
53 \( 1 + (0.412 - 0.910i)T \)
59 \( 1 + (-0.729 + 0.683i)T \)
61 \( 1 + (0.352 - 0.935i)T \)
67 \( 1 + (0.986 - 0.162i)T \)
71 \( 1 + (0.195 + 0.980i)T \)
73 \( 1 + (0.659 - 0.751i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.881 - 0.471i)T \)
89 \( 1 + (0.997 + 0.0654i)T \)
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

−20.29775539073607353338943108865, −19.89982928402875353491442378058, −19.20449515918994663008073172346, −18.288050947618843541981503741988, −17.24236116581255129018358420382, −16.578747832896649232961394216168, −16.234367574817317310783049282378, −15.27253835821915151154149630366, −14.61664655215239449903624276960, −14.211282230954792659613491493183, −12.739644312375141106291676397694, −12.19803881919016834219732958515, −11.359415222717243460138396963099, −10.86742814508221886646396214257, −9.85138200879199399543001258722, −9.19181743838221776579687210119, −8.42775175788622746084173144285, −7.63337979668471580260379887857, −6.65008562368533267095790682237, −5.70224379718802012573261693437, −4.760559686325957688185930521957, −4.21414886834725671288626951850, −3.393340507663009558185988636, −2.55369417298937908773828682957, −0.9148533183649330337552692185, 0.266891661164384024603518662354, 1.60671996246402563925882447377, 2.32812569978425455062344165844, 3.56926533127883271128541050356, 4.20774519921749406782306309218, 5.36010814611675725912290428575, 6.29335360070609275035687040864, 7.018812528057100834603941827255, 7.658950650415649597244524811235, 8.29552311196687542545556054054, 9.258378647026826269984613349995, 10.21260518461889460770607405849, 11.30316940267979114430921069858, 11.721518964220876814953302335058, 12.45171294932084221156825136744, 13.014733638876865082754795071100, 14.100566615274613944729526217874, 14.82415921183932948760277064539, 15.203034797280155990963076922921, 16.617735594517743871322659585927, 16.94677948474764181342216599746, 17.71210861701597647802512816509, 18.77640081736728618140039653721, 19.09218266081396773620548103298, 19.832103988842676145729817710968

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