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Properties

Label 2-1800-40.29-c1-0-25
Degree $2$
Conductor $1800$
Sign $0.971 + 0.237i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + (−0.796 − 1.16i)2-s + (−0.731 + 1.86i)4-s − 4.72i·7-s + (2.75 − 0.627i)8-s + 3.93i·11-s + 3.46·13-s + (−5.51 + 3.76i)14-s + (−2.93 − 2.72i)16-s + 3.51i·17-s + 5.44i·19-s + (4.59 − 3.13i)22-s + 7.11i·23-s + (−2.76 − 4.05i)26-s + (8.79 + 3.45i)28-s − 3.66i·29-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (−0.365 + 0.930i)4-s − 1.78i·7-s + (0.975 − 0.221i)8-s + 1.18i·11-s + 0.961·13-s + (−1.47 + 1.00i)14-s + (−0.732 − 0.680i)16-s + 0.852i·17-s + 1.24i·19-s + (0.979 − 0.667i)22-s + 1.48i·23-s + (−0.541 − 0.794i)26-s + (1.66 + 0.652i)28-s − 0.681i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.971 + 0.237i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.971 + 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.178929869\)
\(L(\frac12)\) \(\approx\) \(1.178929869\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.796 + 1.16i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4.72iT - 7T^{2} \)
11 \( 1 - 3.93iT - 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3.51iT - 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 - 7.11iT - 23T^{2} \)
29 \( 1 + 3.66iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + 0.414T + 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 + 5.34T + 43T^{2} \)
47 \( 1 + 0.925iT - 47T^{2} \)
53 \( 1 + 0.233T + 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 + 0.118iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 0.563iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 8.88T + 89T^{2} \)
97 \( 1 + 7.27iT - 97T^{2} \)
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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

−9.555338799631949698829240651876, −8.399776377349842803802959713250, −7.77162493431677685764499776807, −7.18058010914193926745505462890, −6.22688488292730318165114114433, −4.79525343525995676192277404880, −3.89739121244765044378306485630, −3.54856771272614023100134391819, −1.87268037531085949198677485609, −1.09443989955007071304048175028, 0.64920949423828761517475783172, 2.22949077969884033093103454532, 3.22873907447670781498883868208, 4.83445681971555681884347906907, 5.35149048141535122326279706575, 6.35705288856683856315250967681, 6.60812115064304519209171784431, 8.041886891995796137249089978831, 8.608754874832364168020478317735, 8.960268774849216829928540890070

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