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Properties

Label 2-1045-209.208-c1-0-44
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  − 0.863·2-s − 0.924i·3-s − 1.25·4-s + 5-s + 0.798i·6-s − 0.285i·7-s + 2.81·8-s + 2.14·9-s − 0.863·10-s + (0.818 − 3.21i)11-s + 1.15i·12-s − 0.419·13-s + 0.246i·14-s − 0.924i·15-s + 0.0824·16-s + 2.73i·17-s + ⋯
L(s)  = 1  − 0.610·2-s − 0.533i·3-s − 0.627·4-s + 0.447·5-s + 0.325i·6-s − 0.107i·7-s + 0.993·8-s + 0.715·9-s − 0.273·10-s + (0.246 − 0.969i)11-s + 0.334i·12-s − 0.116·13-s + 0.0658i·14-s − 0.238i·15-s + 0.0206·16-s + 0.663i·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

\(L(1)\) \(\approx\) \(1.145845646\)
\(L(\frac12)\) \(\approx\) \(1.145845646\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.818 + 3.21i)T \)
19 \( 1 + (2.97 - 3.18i)T \)
good2 \( 1 + 0.863T + 2T^{2} \)
3 \( 1 + 0.924iT - 3T^{2} \)
7 \( 1 + 0.285iT - 7T^{2} \)
13 \( 1 + 0.419T + 13T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
23 \( 1 - 7.54T + 23T^{2} \)
29 \( 1 + 2.67T + 29T^{2} \)
31 \( 1 - 3.45iT - 31T^{2} \)
37 \( 1 + 1.21iT - 37T^{2} \)
41 \( 1 - 9.74T + 41T^{2} \)
43 \( 1 + 0.0617iT - 43T^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 + 8.40iT - 59T^{2} \)
61 \( 1 + 2.63iT - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 2.61iT - 71T^{2} \)
73 \( 1 + 7.15iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 14.5iT - 83T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 - 19.2iT - 97T^{2} \)
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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

−9.664536163098814712427132308957, −8.946068324026769691782402200758, −8.242028690906690388673459009780, −7.41953696016611343081097961242, −6.53463228349574074305531401226, −5.59357643737968128573385272721, −4.50982423928998987446827649412, −3.51524918127372356351689065873, −1.87944404367559223347097060810, −0.837492957887982089064206576734, 1.14345209951595162348550385325, 2.55663262776466015642799809762, 4.17124082984977667999912345762, 4.64119760149597280484689294008, 5.60688429473384771155987070178, 7.05047228542274708905707053601, 7.44263413464449807181616900272, 8.867136264846223915747870011107, 9.201033325736490847910820245530, 9.900358745320685795778130106773

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