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Properties

Label 2-1045-5.4-c1-0-69
Degree $2$
Conductor $1045$
Sign $0.502 + 0.864i$
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  + 1.22i·2-s − 1.51i·3-s + 0.500·4-s + (−1.12 − 1.93i)5-s + 1.85·6-s − 0.966i·7-s + 3.06i·8-s + 0.706·9-s + (2.36 − 1.37i)10-s + 11-s − 0.757i·12-s − 3.72i·13-s + 1.18·14-s + (−2.92 + 1.70i)15-s − 2.74·16-s − 2.85i·17-s + ⋯
L(s)  = 1  + 0.865i·2-s − 0.874i·3-s + 0.250·4-s + (−0.502 − 0.864i)5-s + 0.757·6-s − 0.365i·7-s + 1.08i·8-s + 0.235·9-s + (0.748 − 0.435i)10-s + 0.301·11-s − 0.218i·12-s − 1.03i·13-s + 0.316·14-s + (−0.755 + 0.439i)15-s − 0.687·16-s − 0.692i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.502 + 0.864i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.502 + 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.643349089\)
\(L(\frac12)\) \(\approx\) \(1.643349089\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.12 + 1.93i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 1.22iT - 2T^{2} \)
3 \( 1 + 1.51iT - 3T^{2} \)
7 \( 1 + 0.966iT - 7T^{2} \)
13 \( 1 + 3.72iT - 13T^{2} \)
17 \( 1 + 2.85iT - 17T^{2} \)
23 \( 1 + 1.26iT - 23T^{2} \)
29 \( 1 + 2.66T + 29T^{2} \)
31 \( 1 + 0.0694T + 31T^{2} \)
37 \( 1 - 0.887iT - 37T^{2} \)
41 \( 1 + 7.73T + 41T^{2} \)
43 \( 1 - 5.48iT - 43T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 9.57T + 61T^{2} \)
67 \( 1 + 13.5iT - 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 2.93iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + 9.46T + 89T^{2} \)
97 \( 1 + 2.98iT - 97T^{2} \)
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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

−9.628602026320642021522558232687, −8.460636748051758024995976820707, −8.000354337268565540350996421461, −7.21445093733560583642148326185, −6.70465262688936801974864424807, −5.57658931960708509163637282156, −4.86213198307383818865917072010, −3.55202337714188640498890556196, −2.05912090100821806503347186981, −0.75459201500841754253879200373, 1.66881963317342119357951558857, 2.82524088090801619455121073754, 3.81792108974430357566866413528, 4.28236119648373716153044581117, 5.77511982581032477293307672076, 6.80845079724665279697978928340, 7.38055971522031492602654892664, 8.717239255830818266319122242826, 9.549385148973400346881466789075, 10.22692979671892820072756927702

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