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Properties

Label 2-1045-209.208-c1-0-38
Degree $2$
Conductor $1045$
Sign $0.518 + 0.855i$
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  + 0.208·2-s − 0.469i·3-s − 1.95·4-s − 5-s − 0.0976i·6-s + 2.46i·7-s − 0.823·8-s + 2.77·9-s − 0.208·10-s + (−1.36 − 3.02i)11-s + 0.917i·12-s − 4.03·13-s + 0.513i·14-s + 0.469i·15-s + 3.74·16-s + 0.922i·17-s + ⋯
L(s)  = 1  + 0.147·2-s − 0.270i·3-s − 0.978·4-s − 0.447·5-s − 0.0398i·6-s + 0.932i·7-s − 0.291·8-s + 0.926·9-s − 0.0658·10-s + (−0.412 − 0.911i)11-s + 0.264i·12-s − 1.11·13-s + 0.137i·14-s + 0.121i·15-s + 0.935·16-s + 0.223i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.518 + 0.855i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.518 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.074815484\)
\(L(\frac12)\) \(\approx\) \(1.074815484\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 + 3.02i)T \)
19 \( 1 + (-4.32 + 0.522i)T \)
good2 \( 1 - 0.208T + 2T^{2} \)
3 \( 1 + 0.469iT - 3T^{2} \)
7 \( 1 - 2.46iT - 7T^{2} \)
13 \( 1 + 4.03T + 13T^{2} \)
17 \( 1 - 0.922iT - 17T^{2} \)
23 \( 1 - 0.0784T + 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 8.68iT - 31T^{2} \)
37 \( 1 + 8.63iT - 37T^{2} \)
41 \( 1 - 8.99T + 41T^{2} \)
43 \( 1 - 4.97iT - 43T^{2} \)
47 \( 1 - 1.86T + 47T^{2} \)
53 \( 1 + 2.85iT - 53T^{2} \)
59 \( 1 + 7.93iT - 59T^{2} \)
61 \( 1 + 1.44iT - 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 + 6.57iT - 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + 2.57T + 79T^{2} \)
83 \( 1 - 5.78iT - 83T^{2} \)
89 \( 1 - 7.54iT - 89T^{2} \)
97 \( 1 + 8.63iT - 97T^{2} \)
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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

−9.509476519785155257874322113383, −9.148479872938601290394342762776, −7.936600563933591012346764633279, −7.62281557684700558871420348188, −6.23182257928394261576160283690, −5.38184625379518001330869175709, −4.58774166504434600595258538080, −3.56832426447938841865581147194, −2.40302484927842324055091713138, −0.59482754315260367087186242941, 1.10255310696696239824899307417, 2.98186868790633775760268223409, 4.16848171150945427497423433939, 4.60280289082744940697233634493, 5.40823506526047662897590224398, 7.17712269827333344541306465459, 7.28161012711179062865084148478, 8.398170769585445288685149466871, 9.447309249718290352163389712230, 10.07727706856705548127082881392

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