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Properties

Label 2-1045-209.208-c1-0-23
Degree $2$
Conductor $1045$
Sign $-0.855 - 0.518i$
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  + 2.08·2-s + 2.80i·3-s + 2.35·4-s − 5-s + 5.85i·6-s + 0.950i·7-s + 0.737·8-s − 4.88·9-s − 2.08·10-s + (0.932 + 3.18i)11-s + 6.60i·12-s − 3.48·13-s + 1.98i·14-s − 2.80i·15-s − 3.16·16-s + 1.26i·17-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.62i·3-s + 1.17·4-s − 0.447·5-s + 2.39i·6-s + 0.359i·7-s + 0.260·8-s − 1.62·9-s − 0.659·10-s + (0.281 + 0.959i)11-s + 1.90i·12-s − 0.967·13-s + 0.529i·14-s − 0.725i·15-s − 0.791·16-s + 0.306i·17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.684817939\)
\(L(\frac12)\) \(\approx\) \(2.684817939\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.932 - 3.18i)T \)
19 \( 1 + (-3.21 + 2.94i)T \)
good2 \( 1 - 2.08T + 2T^{2} \)
3 \( 1 - 2.80iT - 3T^{2} \)
7 \( 1 - 0.950iT - 7T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 - 1.26iT - 17T^{2} \)
23 \( 1 - 4.91T + 23T^{2} \)
29 \( 1 + 4.16T + 29T^{2} \)
31 \( 1 - 6.79iT - 31T^{2} \)
37 \( 1 - 3.59iT - 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 - 9.25T + 47T^{2} \)
53 \( 1 - 3.95iT - 53T^{2} \)
59 \( 1 - 3.82iT - 59T^{2} \)
61 \( 1 + 3.98iT - 61T^{2} \)
67 \( 1 + 7.67iT - 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 6.81iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 - 4.45iT - 89T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−10.46678552991423829924857382354, −9.297991127409725514620140254528, −9.102921866841145594863065364295, −7.54435537890014037081035548162, −6.64358156002097589675436082512, −5.33906321349012389843634580658, −4.97262913668968239186594163770, −4.24052362090467863398371496703, −3.42710351339842268205216448612, −2.54553496290770867820274694799, 0.72948155125409971201677758403, 2.29629409145149601667868712512, 3.21645397946669769845515896072, 4.21856409982747932172419537938, 5.47102256511360547121148855305, 5.99802938302108857738382221878, 7.14450700658205721966217311032, 7.35995765835927797194560139604, 8.440665203103227104569202476555, 9.475112738027680199500267815447

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