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Properties

Label 1-1792-1792.293-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.844 - 0.534i$
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  + (0.471 + 0.881i)3-s + (0.773 − 0.634i)5-s + (−0.555 + 0.831i)9-s + (0.290 + 0.956i)11-s + (0.634 − 0.773i)13-s + (0.923 + 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.995 + 0.0980i)19-s + (−0.980 + 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.995 − 0.0980i)27-s + (0.956 + 0.290i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.0980 − 0.995i)37-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)3-s + (0.773 − 0.634i)5-s + (−0.555 + 0.831i)9-s + (0.290 + 0.956i)11-s + (0.634 − 0.773i)13-s + (0.923 + 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.995 + 0.0980i)19-s + (−0.980 + 0.195i)23-s + (0.195 − 0.980i)25-s + (−0.995 − 0.0980i)27-s + (0.956 + 0.290i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.0980 − 0.995i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.844 - 0.534i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.844 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.358160723 - 0.6838539374i\)
\(L(\frac12)\) \(\approx\) \(2.358160723 - 0.6838539374i\)
\(L(1)\) \(\approx\) \(1.323162094 + 0.1635081112i\)
\(L(1)\) \(\approx\) \(1.323162094 + 0.1635081112i\)

Euler product

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

−19.90095853187798365560934537943, −19.26499980402088749808344308299, −18.59777449949830486270029727333, −18.02060457700279281641714222508, −17.38473243531634744638271306508, −16.47473132556139071540278545423, −15.62245243844848270552282051358, −14.472460165841989127707117810424, −14.221425293734666577378587195987, −13.39427080668158543238462682151, −12.966419401020735799161250744883, −11.74159235016499986847976022839, −11.24392008212926618301316827823, −10.32705510343557037775290728618, −9.31775817713209269639755946925, −8.687042540369215324849251608991, −8.008763939320210907253346846485, −6.72061657658339167561915832834, −6.52284123772013365837366406528, −5.79213991949930240138702493350, −4.431957391257866093135333491852, −3.41917370886834946061786598599, −2.58074547095360375231866029725, −1.860686321720265332559463512342, −0.93055782779999032958688008323, 0.42187281497504506869718885936, 1.93885271796333138648291689738, 2.31122418415714792682311361312, 3.78551528355709372122294947062, 4.24100025563934083745101871061, 5.23268096284150324841838162611, 5.87556668826776448572585838615, 6.88132779478850563561646948114, 8.08613696948641888308924798649, 8.71873906074071347362184974417, 9.29802886480243703499903010717, 10.28750215984589602913107057538, 10.51070635636387896922276571836, 11.70378674652925316791255689369, 12.72492192440443191345169131217, 13.24100358828294717422658895971, 14.10667500545184647432916222872, 14.80202035519265738153445941100, 15.579062804218453813408387525257, 16.130977975220996301757795755818, 17.11806336044027156152287544702, 17.539948489437822582404575871349, 18.34538429689215124705809920598, 19.60316475874138853092864231306, 20.08173472196484960840028865820

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