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Properties

Label 2-1045-5.4-c1-0-43
Degree $2$
Conductor $1045$
Sign $0.897 - 0.441i$
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  + 2.04i·2-s − 1.84i·3-s − 2.16·4-s + (−2.00 + 0.987i)5-s + 3.75·6-s − 2.14i·7-s − 0.342i·8-s − 0.388·9-s + (−2.01 − 4.09i)10-s + 11-s + 3.98i·12-s + 3.36i·13-s + 4.37·14-s + (1.81 + 3.69i)15-s − 3.63·16-s − 6.44i·17-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.06i·3-s − 1.08·4-s + (−0.897 + 0.441i)5-s + 1.53·6-s − 0.809i·7-s − 0.120i·8-s − 0.129·9-s + (−0.637 − 1.29i)10-s + 0.301·11-s + 1.15i·12-s + 0.932i·13-s + 1.16·14-s + (0.469 + 0.953i)15-s − 0.909·16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.897 - 0.441i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.897 - 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.340006941\)
\(L(\frac12)\) \(\approx\) \(1.340006941\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.00 - 0.987i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 2.04iT - 2T^{2} \)
3 \( 1 + 1.84iT - 3T^{2} \)
7 \( 1 + 2.14iT - 7T^{2} \)
13 \( 1 - 3.36iT - 13T^{2} \)
17 \( 1 + 6.44iT - 17T^{2} \)
23 \( 1 - 0.280iT - 23T^{2} \)
29 \( 1 - 9.19T + 29T^{2} \)
31 \( 1 - 0.844T + 31T^{2} \)
37 \( 1 - 3.36iT - 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 9.58iT - 43T^{2} \)
47 \( 1 - 6.66iT - 47T^{2} \)
53 \( 1 + 5.21iT - 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 6.13T + 61T^{2} \)
67 \( 1 + 7.00iT - 67T^{2} \)
71 \( 1 + 4.09T + 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 - 7.51T + 79T^{2} \)
83 \( 1 + 6.85iT - 83T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 - 17.7iT - 97T^{2} \)
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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

−9.726696076551460916206442398591, −8.712882369552730346572726297389, −7.87811722570276819692779938293, −7.22918603446672031078492910978, −6.92413793695797027321935475649, −6.24258230435701589837531389598, −4.85106662832775890653106078354, −4.10167300127007812540600294706, −2.57168015454663392904307737697, −0.77653135058547757543838673796, 1.10365675582976700726427260139, 2.64627815138339205904256534286, 3.60458759500122084887204773054, 4.23238084322140517900167963940, 5.06076681843443229914835842195, 6.23636962600725136776782437187, 7.70742749877584200957705710749, 8.657709812707175080870430290967, 9.191871629764451536965435950820, 10.15373870691132065019406891939

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