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Properties

Label 2-1045-209.208-c1-0-78
Degree $2$
Conductor $1045$
Sign $-0.540 - 0.841i$
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  + 0.136·2-s − 2.54i·3-s − 1.98·4-s + 5-s − 0.345i·6-s − 4.55i·7-s − 0.541·8-s − 3.46·9-s + 0.136·10-s + (−3.25 + 0.634i)11-s + 5.03i·12-s − 5.85·13-s − 0.619i·14-s − 2.54i·15-s + 3.88·16-s + 1.74i·17-s + ⋯
L(s)  = 1  + 0.0961·2-s − 1.46i·3-s − 0.990·4-s + 0.447·5-s − 0.141i·6-s − 1.72i·7-s − 0.191·8-s − 1.15·9-s + 0.0430·10-s + (−0.981 + 0.191i)11-s + 1.45i·12-s − 1.62·13-s − 0.165i·14-s − 0.656i·15-s + 0.972·16-s + 0.424i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.540 - 0.841i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5263964617\)
\(L(\frac12)\) \(\approx\) \(0.5263964617\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 - 0.634i)T \)
19 \( 1 + (-1.61 - 4.04i)T \)
good2 \( 1 - 0.136T + 2T^{2} \)
3 \( 1 + 2.54iT - 3T^{2} \)
7 \( 1 + 4.55iT - 7T^{2} \)
13 \( 1 + 5.85T + 13T^{2} \)
17 \( 1 - 1.74iT - 17T^{2} \)
23 \( 1 - 5.13T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 + 8.20iT - 31T^{2} \)
37 \( 1 + 0.954iT - 37T^{2} \)
41 \( 1 + 5.61T + 41T^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 - 5.12T + 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 - 8.16iT - 59T^{2} \)
61 \( 1 + 3.33iT - 61T^{2} \)
67 \( 1 + 8.57iT - 67T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 - 1.37iT - 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 + 4.06iT - 89T^{2} \)
97 \( 1 - 1.46iT - 97T^{2} \)
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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

−9.523424157325063063881786152906, −8.206426624504525022343287228126, −7.52413467532396953812026115616, −7.20435166883597780923516617211, −5.99807975856201136440707907043, −5.04394438039995539490298206382, −4.11962513427217256121515851022, −2.75923822640736657277764820697, −1.40311108107900446261991400352, −0.23993631800971351402150779949, 2.55888935596757501702768339552, 3.22688749558087882213341278451, 4.84287391788284043028868469710, 5.05939157389515216711920513265, 5.58531151280039528608630133286, 7.14009597954385865070605254802, 8.560103602112473193134790047615, 8.910172082752992028555593167115, 9.706156041558847700391293548290, 10.05073785025830379827768202235

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