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Properties

Label 2-1045-5.4-c1-0-6
Degree $2$
Conductor $1045$
Sign $0.469 - 0.883i$
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  − 2.39i·2-s + 3.31i·3-s − 3.71·4-s + (1.04 − 1.97i)5-s + 7.92·6-s + 0.907i·7-s + 4.10i·8-s − 7.98·9-s + (−4.72 − 2.50i)10-s + 11-s − 12.3i·12-s + 4.80i·13-s + 2.17·14-s + (6.54 + 3.47i)15-s + 2.38·16-s + 3.57i·17-s + ⋯
L(s)  = 1  − 1.69i·2-s + 1.91i·3-s − 1.85·4-s + (0.469 − 0.883i)5-s + 3.23·6-s + 0.343i·7-s + 1.45i·8-s − 2.66·9-s + (−1.49 − 0.793i)10-s + 0.301·11-s − 3.55i·12-s + 1.33i·13-s + 0.580·14-s + (1.68 + 0.897i)15-s + 0.596·16-s + 0.868i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.469 - 0.883i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9486574875\)
\(L(\frac12)\) \(\approx\) \(0.9486574875\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.04 + 1.97i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 2.39iT - 2T^{2} \)
3 \( 1 - 3.31iT - 3T^{2} \)
7 \( 1 - 0.907iT - 7T^{2} \)
13 \( 1 - 4.80iT - 13T^{2} \)
17 \( 1 - 3.57iT - 17T^{2} \)
23 \( 1 + 0.159iT - 23T^{2} \)
29 \( 1 - 2.96T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 - 8.00iT - 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 - 2.33iT - 43T^{2} \)
47 \( 1 - 6.15iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 - 0.603T + 61T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 + 8.43iT - 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 - 4.61iT - 83T^{2} \)
89 \( 1 - 7.69T + 89T^{2} \)
97 \( 1 - 1.90iT - 97T^{2} \)
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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

−10.14798016007449867421648466326, −9.336880001946379539701437262640, −9.061625836256915249490163705576, −8.405014558446068355081514472614, −6.20028793690589604640130729901, −5.19917096451199082010055709321, −4.39175518735313837302125023304, −3.96881358748314164234140487415, −2.84164879042912756642554711588, −1.65559189203471447979003829728, 0.41982803423896815484280214117, 2.13323071134290914511490235455, 3.37793740413901582679555925967, 5.31822452124378904946961842774, 5.81726653825711588445178940158, 6.68732190217297300900042440606, 7.16354825904437933821700653645, 7.67406929556258990911301163155, 8.456626527735236158916260615639, 9.324928300501405361724007775326

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