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Properties

Label 2-1040-65.64-c1-0-8
Degree $2$
Conductor $1040$
Sign $0.944 - 0.328i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  − 3.22i·3-s + (−0.589 + 2.15i)5-s + 1.65·7-s − 7.41·9-s + 4.44i·11-s + (2.04 + 2.97i)13-s + (6.95 + 1.90i)15-s + 5.33i·17-s − 0.472i·19-s − 5.33i·21-s + 1.08i·23-s + (−4.30 − 2.54i)25-s + 14.2i·27-s + 5.50·29-s + 5.47i·31-s + ⋯
L(s)  = 1  − 1.86i·3-s + (−0.263 + 0.964i)5-s + 0.625·7-s − 2.47·9-s + 1.34i·11-s + (0.566 + 0.824i)13-s + (1.79 + 0.491i)15-s + 1.29i·17-s − 0.108i·19-s − 1.16i·21-s + 0.226i·23-s + (−0.860 − 0.508i)25-s + 2.73i·27-s + 1.02·29-s + 0.982i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.328879042\)
\(L(\frac12)\) \(\approx\) \(1.328879042\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.589 - 2.15i)T \)
13 \( 1 + (-2.04 - 2.97i)T \)
good3 \( 1 + 3.22iT - 3T^{2} \)
7 \( 1 - 1.65T + 7T^{2} \)
11 \( 1 - 4.44iT - 11T^{2} \)
17 \( 1 - 5.33iT - 17T^{2} \)
19 \( 1 + 0.472iT - 19T^{2} \)
23 \( 1 - 1.08iT - 23T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 - 5.47iT - 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 - 10.4iT - 41T^{2} \)
43 \( 1 + 8.14iT - 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 + 4.14iT - 53T^{2} \)
59 \( 1 + 2.52iT - 59T^{2} \)
61 \( 1 - 0.160T + 61T^{2} \)
67 \( 1 - 8.26T + 67T^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 8.65T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 7.61iT - 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−10.13029746176972285752308655900, −8.772658533520290021458086931619, −8.073236172251528195380551134822, −7.38207124575019608226052363995, −6.66020555474477779717803887473, −6.22375003101911567128746667949, −4.83011651140278480211774886596, −3.45400299421533641320898490759, −2.16123510338490939759453734421, −1.53586165455080469020649513805, 0.62203956335964590264804857224, 2.90203556056705662347888427899, 3.78036800531270663367558229718, 4.66743266273655452047389812007, 5.31455445533455047585068567929, 5.99599339105435652854997543126, 7.81116059971123377507442843398, 8.570603098265306893752523821784, 8.960773863172736911006443796673, 9.883988359720766331932031242737

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