Properties

Label 2-1045-1.1-c5-0-209
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.84·2-s + 26.7·3-s − 17.2·4-s + 25·5-s − 103.·6-s + 117.·7-s + 189.·8-s + 475.·9-s − 96.1·10-s + 121·11-s − 461.·12-s + 571.·13-s − 453.·14-s + 669.·15-s − 176.·16-s + 1.32e3·17-s − 1.82e3·18-s + 361·19-s − 430.·20-s + 3.15e3·21-s − 465.·22-s + 1.79e3·23-s + 5.07e3·24-s + 625·25-s − 2.19e3·26-s + 6.22e3·27-s − 2.02e3·28-s + ⋯
L(s)  = 1  − 0.679·2-s + 1.71·3-s − 0.538·4-s + 0.447·5-s − 1.16·6-s + 0.909·7-s + 1.04·8-s + 1.95·9-s − 0.303·10-s + 0.301·11-s − 0.925·12-s + 0.937·13-s − 0.617·14-s + 0.768·15-s − 0.172·16-s + 1.11·17-s − 1.32·18-s + 0.229·19-s − 0.240·20-s + 1.56·21-s − 0.204·22-s + 0.706·23-s + 1.79·24-s + 0.200·25-s − 0.636·26-s + 1.64·27-s − 0.489·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.369185498\)
\(L(\frac12)\) \(\approx\) \(4.369185498\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25T \)
11 \( 1 - 121T \)
19 \( 1 - 361T \)
good2 \( 1 + 3.84T + 32T^{2} \)
3 \( 1 - 26.7T + 243T^{2} \)
7 \( 1 - 117.T + 1.68e4T^{2} \)
13 \( 1 - 571.T + 3.71e5T^{2} \)
17 \( 1 - 1.32e3T + 1.41e6T^{2} \)
23 \( 1 - 1.79e3T + 6.43e6T^{2} \)
29 \( 1 + 882.T + 2.05e7T^{2} \)
31 \( 1 - 1.34e3T + 2.86e7T^{2} \)
37 \( 1 - 5.58e3T + 6.93e7T^{2} \)
41 \( 1 - 4.28e3T + 1.15e8T^{2} \)
43 \( 1 + 694.T + 1.47e8T^{2} \)
47 \( 1 - 1.77e4T + 2.29e8T^{2} \)
53 \( 1 - 1.62e4T + 4.18e8T^{2} \)
59 \( 1 + 1.90e4T + 7.14e8T^{2} \)
61 \( 1 + 1.49e4T + 8.44e8T^{2} \)
67 \( 1 - 3.51e4T + 1.35e9T^{2} \)
71 \( 1 + 7.92e4T + 1.80e9T^{2} \)
73 \( 1 + 4.53e4T + 2.07e9T^{2} \)
79 \( 1 + 1.69e4T + 3.07e9T^{2} \)
83 \( 1 + 1.03e5T + 3.93e9T^{2} \)
89 \( 1 + 2.05e4T + 5.58e9T^{2} \)
97 \( 1 + 8.98e3T + 8.58e9T^{2} \)
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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

−8.976251934761381339985386703001, −8.561652113265795226804475965893, −7.82012852006562451978374437370, −7.24261484055006728472865251548, −5.74215579862094396515710122966, −4.59806903407048357551957420716, −3.79234623739861797258446798533, −2.79282474340376147589275429086, −1.57181930843155997131076474685, −1.08811075737375676974004054973, 1.08811075737375676974004054973, 1.57181930843155997131076474685, 2.79282474340376147589275429086, 3.79234623739861797258446798533, 4.59806903407048357551957420716, 5.74215579862094396515710122966, 7.24261484055006728472865251548, 7.82012852006562451978374437370, 8.561652113265795226804475965893, 8.976251934761381339985386703001

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