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Properties

Label 2-1045-5.4-c1-0-7
Degree $2$
Conductor $1045$
Sign $0.141 + 0.989i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.77i·2-s − 2.74i·3-s − 5.69·4-s + (0.316 + 2.21i)5-s − 7.61·6-s + 3.20i·7-s + 10.2i·8-s − 4.53·9-s + (6.14 − 0.877i)10-s + 11-s + 15.6i·12-s − 3.87i·13-s + 8.90·14-s + (6.07 − 0.867i)15-s + 17.0·16-s + 5.86i·17-s + ⋯
L(s)  = 1  − 1.96i·2-s − 1.58i·3-s − 2.84·4-s + (0.141 + 0.989i)5-s − 3.10·6-s + 1.21i·7-s + 3.62i·8-s − 1.51·9-s + (1.94 − 0.277i)10-s + 0.301·11-s + 4.51i·12-s − 1.07i·13-s + 2.37·14-s + (1.56 − 0.224i)15-s + 4.26·16-s + 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.141 + 0.989i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9316260969\)
\(L(\frac12)\) \(\approx\) \(0.9316260969\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.316 - 2.21i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 2.77iT - 2T^{2} \)
3 \( 1 + 2.74iT - 3T^{2} \)
7 \( 1 - 3.20iT - 7T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 - 5.86iT - 17T^{2} \)
23 \( 1 - 6.11iT - 23T^{2} \)
29 \( 1 + 6.40T + 29T^{2} \)
31 \( 1 - 1.66T + 31T^{2} \)
37 \( 1 - 0.251iT - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 2.17iT - 43T^{2} \)
47 \( 1 - 2.68iT - 47T^{2} \)
53 \( 1 - 8.89iT - 53T^{2} \)
59 \( 1 - 4.09T + 59T^{2} \)
61 \( 1 - 4.52T + 61T^{2} \)
67 \( 1 - 2.61iT - 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 - 2.73iT - 73T^{2} \)
79 \( 1 + 8.17T + 79T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 - 1.84T + 89T^{2} \)
97 \( 1 + 2.65iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.925619186945598649234120912222, −9.061524923106299231098709916106, −8.218571840657697107224006594427, −7.53590131559175742437701285677, −5.98411003012178031884331318989, −5.63169663154382212181640886328, −3.85067248715618254547430165690, −2.86066326977913906225739233948, −2.21705361623104329652892895106, −1.33043650968998757852143152869, 0.46944608291777505269262345649, 3.79941048817196330558850924478, 4.38136859220317489065720602311, 4.81819578438730906882717871273, 5.71238140142505478108517908545, 6.75970393456330748924930148100, 7.53445038832881103350860821416, 8.587314976844584548734107756566, 9.170103681524089429672080904466, 9.647675776770639084254742522236

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