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Properties

Label 2-1045-5.4-c1-0-56
Degree $2$
Conductor $1045$
Sign $0.694 - 0.719i$
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  + 2.44i·2-s − 0.661i·3-s − 3.97·4-s + (1.55 − 1.60i)5-s + 1.61·6-s + 2.05i·7-s − 4.82i·8-s + 2.56·9-s + (3.93 + 3.79i)10-s + 11-s + 2.62i·12-s − 7.18i·13-s − 5.01·14-s + (−1.06 − 1.02i)15-s + 3.83·16-s − 3.65i·17-s + ⋯
L(s)  = 1  + 1.72i·2-s − 0.381i·3-s − 1.98·4-s + (0.694 − 0.719i)5-s + 0.659·6-s + 0.775i·7-s − 1.70i·8-s + 0.854·9-s + (1.24 + 1.19i)10-s + 0.301·11-s + 0.758i·12-s − 1.99i·13-s − 1.34·14-s + (−0.274 − 0.265i)15-s + 0.959·16-s − 0.886i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.694 - 0.719i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665126059\)
\(L(\frac12)\) \(\approx\) \(1.665126059\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.55 + 1.60i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 - 2.44iT - 2T^{2} \)
3 \( 1 + 0.661iT - 3T^{2} \)
7 \( 1 - 2.05iT - 7T^{2} \)
13 \( 1 + 7.18iT - 13T^{2} \)
17 \( 1 + 3.65iT - 17T^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 - 0.355T + 29T^{2} \)
31 \( 1 - 3.81T + 31T^{2} \)
37 \( 1 - 3.78iT - 37T^{2} \)
41 \( 1 + 6.80T + 41T^{2} \)
43 \( 1 + 2.39iT - 43T^{2} \)
47 \( 1 - 9.86iT - 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 - 7.64T + 61T^{2} \)
67 \( 1 - 11.9iT - 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 + 3.96T + 79T^{2} \)
83 \( 1 - 9.80iT - 83T^{2} \)
89 \( 1 - 8.04T + 89T^{2} \)
97 \( 1 - 5.99iT - 97T^{2} \)
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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

−9.742522986796661242401007043587, −8.792907498847062221834210983545, −8.301039231880267016024073282654, −7.48461973863426967448481461270, −6.57690451513831936416047753286, −5.86321274223365305335082784928, −5.17871146012734722479799386972, −4.42054125430950920103611055758, −2.63001717894696725701097971603, −0.841958425334379819770925393688, 1.47706979815134711862218007520, 2.11956204575736303574375776137, 3.68467171681145967744228795225, 3.96724230009170524959197541031, 5.04077892582326707376294668036, 6.53471575983996884536268837883, 7.16651809421775825361320254257, 8.692701709840893237514369516506, 9.465686550512396786769865912520, 10.08711228589152594538331079585

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