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Properties

Label 1-1045-1045.614-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.177 - 0.984i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.719 − 0.694i)2-s + (−0.848 + 0.529i)3-s + (0.0348 − 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.669 + 0.743i)7-s + (−0.669 − 0.743i)8-s + (0.438 − 0.898i)9-s + (0.5 + 0.866i)12-s + (0.997 − 0.0697i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (−0.438 − 0.898i)17-s + (−0.309 − 0.951i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.961 + 0.275i)24-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (−0.848 + 0.529i)3-s + (0.0348 − 0.999i)4-s + (−0.241 + 0.970i)6-s + (−0.669 + 0.743i)7-s + (−0.669 − 0.743i)8-s + (0.438 − 0.898i)9-s + (0.5 + 0.866i)12-s + (0.997 − 0.0697i)13-s + (0.0348 + 0.999i)14-s + (−0.997 − 0.0697i)16-s + (−0.438 − 0.898i)17-s + (−0.309 − 0.951i)18-s + (0.173 − 0.984i)21-s + (0.939 + 0.342i)23-s + (0.961 + 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.177 - 0.984i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.177 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8229675101 - 0.9850677680i\)
\(L(\frac12)\) \(\approx\) \(0.8229675101 - 0.9850677680i\)
\(L(1)\) \(\approx\) \(0.9944642763 - 0.4005728246i\)
\(L(1)\) \(\approx\) \(0.9944642763 - 0.4005728246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
3 \( 1 + (-0.848 + 0.529i)T \)
7 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.997 - 0.0697i)T \)
17 \( 1 + (-0.438 - 0.898i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.848 - 0.529i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.990 - 0.139i)T \)
53 \( 1 + (-0.559 - 0.829i)T \)
59 \( 1 + (0.990 - 0.139i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.559 - 0.829i)T \)
73 \( 1 + (0.374 - 0.927i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.719 - 0.694i)T \)
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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

−22.12007262749651718743688115401, −21.15487748401996901026427850405, −20.39990489050733601490998519426, −19.27191256131310398923696315167, −18.54588102464526249043138858340, −17.507424587225595149629701013674, −17.02913599559710303183977385880, −16.30801230926479131099299454820, −15.65488340788510368279632820706, −14.67419341841564500462421104623, −13.621288068069846637088916160508, −13.050798725762364808287465151805, −12.66518382173001961861208505898, −11.371549734353972350009236077627, −10.990411354659659646190774322944, −9.78352108862233322569092006428, −8.55000212934857078486862243129, −7.69663333923083288456225666167, −6.84196546120416584192365486089, −6.267568023012444732536719285195, −5.57635860232122040951636688096, −4.40834035344644375762403556890, −3.78177322988616423632236299592, −2.52135816576384299105911855394, −1.10000049660429005884891745166, 0.54484301238538387472318327828, 1.86756098833496927566855921480, 3.144297367766829324273109787541, 3.73985096661812997082872373789, 4.91338645867275349059745631315, 5.5064910746242096056985190804, 6.30645579281742264607463003700, 7.0720817662988492687652994426, 8.985176309054479335028341996336, 9.31354593583037792929058416230, 10.392882594434142317424669933792, 11.10787187475554236450425967557, 11.67511505291299253733633342152, 12.640843901426975586091773814283, 13.078750746062352796522108385556, 14.20556092881842547986944005921, 15.10969552557487639113893793318, 15.9237544099954215757815295347, 16.19092412460182591724545951823, 17.62455824700004928360436319002, 18.29664383075589606111420135605, 19.01909308845547327713365231398, 19.90579057935613437384266825797, 20.98552847694796670926366405398, 21.197524727978965458268664708035

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