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Properties

Label 2-1045-209.208-c1-0-48
Degree $2$
Conductor $1045$
Sign $0.951 + 0.306i$
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  + 0.743·2-s + 1.99i·3-s − 1.44·4-s − 5-s + 1.48i·6-s − 2.79i·7-s − 2.56·8-s − 0.986·9-s − 0.743·10-s + (3.28 − 0.443i)11-s − 2.89i·12-s − 1.27·13-s − 2.07i·14-s − 1.99i·15-s + 0.990·16-s − 5.64i·17-s + ⋯
L(s)  = 1  + 0.525·2-s + 1.15i·3-s − 0.723·4-s − 0.447·5-s + 0.605i·6-s − 1.05i·7-s − 0.905·8-s − 0.328·9-s − 0.235·10-s + (0.991 − 0.133i)11-s − 0.834i·12-s − 0.353·13-s − 0.555i·14-s − 0.515i·15-s + 0.247·16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.951 + 0.306i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.951 + 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.436498960\)
\(L(\frac12)\) \(\approx\) \(1.436498960\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
11 \( 1 + (-3.28 + 0.443i)T \)
19 \( 1 + (3.93 + 1.87i)T \)
good2 \( 1 - 0.743T + 2T^{2} \)
3 \( 1 - 1.99iT - 3T^{2} \)
7 \( 1 + 2.79iT - 7T^{2} \)
13 \( 1 + 1.27T + 13T^{2} \)
17 \( 1 + 5.64iT - 17T^{2} \)
23 \( 1 - 7.14T + 23T^{2} \)
29 \( 1 - 4.39T + 29T^{2} \)
31 \( 1 + 4.14iT - 31T^{2} \)
37 \( 1 - 3.77iT - 37T^{2} \)
41 \( 1 - 3.60T + 41T^{2} \)
43 \( 1 - 0.546iT - 43T^{2} \)
47 \( 1 + 3.88T + 47T^{2} \)
53 \( 1 + 7.00iT - 53T^{2} \)
59 \( 1 + 7.28iT - 59T^{2} \)
61 \( 1 + 3.12iT - 61T^{2} \)
67 \( 1 - 3.21iT - 67T^{2} \)
71 \( 1 + 6.06iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 - 6.14T + 79T^{2} \)
83 \( 1 + 5.92iT - 83T^{2} \)
89 \( 1 - 9.79iT - 89T^{2} \)
97 \( 1 + 9.33iT - 97T^{2} \)
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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

−9.700241892812864955820957881497, −9.311250954319555086877175316888, −8.446973618681514636565243186024, −7.26581841286996664812917917408, −6.46764456736348388363209130135, −4.96600343927167020167154712809, −4.63646861081559303245560372647, −3.86198686458280812935565936315, −3.08684257105463786696418254811, −0.66123180896495189387316980607, 1.24426812187200859480904372470, 2.56355450436233463308642993033, 3.80353650874942548692030756060, 4.68685307507995054615574426577, 5.83138437095740274087911821240, 6.45149408506764392973005912837, 7.36553830626316091773967498973, 8.602161484286135155356075474837, 8.664356322745255649639417360288, 9.831879592094943974501044351506

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