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Properties

Label 2-1045-5.4-c1-0-61
Degree $2$
Conductor $1045$
Sign $0.708 + 0.706i$
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  − 0.961i·2-s + 0.510i·3-s + 1.07·4-s + (1.58 + 1.57i)5-s + 0.490·6-s − 0.688i·7-s − 2.95i·8-s + 2.73·9-s + (1.51 − 1.52i)10-s + 11-s + 0.548i·12-s − 6.48i·13-s − 0.661·14-s + (−0.805 + 0.807i)15-s − 0.693·16-s + 4.14i·17-s + ⋯
L(s)  = 1  − 0.679i·2-s + 0.294i·3-s + 0.537·4-s + (0.708 + 0.706i)5-s + 0.200·6-s − 0.260i·7-s − 1.04i·8-s + 0.913·9-s + (0.480 − 0.481i)10-s + 0.301·11-s + 0.158i·12-s − 1.79i·13-s − 0.176·14-s + (−0.207 + 0.208i)15-s − 0.173·16-s + 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.708 + 0.706i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.708 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.362852460\)
\(L(\frac12)\) \(\approx\) \(2.362852460\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 - 1.57i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 0.961iT - 2T^{2} \)
3 \( 1 - 0.510iT - 3T^{2} \)
7 \( 1 + 0.688iT - 7T^{2} \)
13 \( 1 + 6.48iT - 13T^{2} \)
17 \( 1 - 4.14iT - 17T^{2} \)
23 \( 1 + 5.52iT - 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 - 6.18T + 31T^{2} \)
37 \( 1 + 0.887iT - 37T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + 2.89iT - 47T^{2} \)
53 \( 1 - 9.37iT - 53T^{2} \)
59 \( 1 + 1.00T + 59T^{2} \)
61 \( 1 + 4.33T + 61T^{2} \)
67 \( 1 + 1.18iT - 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 1.17iT - 73T^{2} \)
79 \( 1 + 1.99T + 79T^{2} \)
83 \( 1 - 0.103iT - 83T^{2} \)
89 \( 1 + 9.70T + 89T^{2} \)
97 \( 1 - 0.708iT - 97T^{2} \)
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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

−10.05396279906579103143980427002, −9.446621115636852236498945729568, −8.068299932195148258372259267016, −7.25353547602103765662521499430, −6.39530389343359248471067062840, −5.67522137418086430803530492890, −4.24387485632727282457589449253, −3.34235972604011684372836949355, −2.40645915445757843137769993484, −1.23522952129720173318236700077, 1.55418994562822611800591790810, 2.25158914585441574516664505070, 4.02660050906700245086958707966, 5.05377035917186054177653486001, 5.87228117699895478243755345815, 6.80855661685750145760839007626, 7.22075618958905700941086371840, 8.285872315090936505868958517448, 9.297168718384192860724161157539, 9.617693282165874603603861936153

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