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Properties

Label 2-1800-5.4-c3-0-3
Degree $2$
Conductor $1800$
Sign $-0.894 + 0.447i$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 12i·7-s − 64·11-s + 58i·13-s + 32i·17-s + 136·19-s + 128i·23-s − 144·29-s + 20·31-s + 18i·37-s + 288·41-s − 200i·43-s + 384i·47-s + 199·49-s − 496i·53-s − 128·59-s + ⋯
L(s)  = 1  + 0.647i·7-s − 1.75·11-s + 1.23i·13-s + 0.456i·17-s + 1.64·19-s + 1.16i·23-s − 0.922·29-s + 0.115·31-s + 0.0799i·37-s + 1.09·41-s − 0.709i·43-s + 1.19i·47-s + 0.580·49-s − 1.28i·53-s − 0.282·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3932152873\)
\(L(\frac12)\) \(\approx\) \(0.3932152873\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 12iT - 343T^{2} \)
11 \( 1 + 64T + 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
17 \( 1 - 32iT - 4.91e3T^{2} \)
19 \( 1 - 136T + 6.85e3T^{2} \)
23 \( 1 - 128iT - 1.21e4T^{2} \)
29 \( 1 + 144T + 2.43e4T^{2} \)
31 \( 1 - 20T + 2.97e4T^{2} \)
37 \( 1 - 18iT - 5.06e4T^{2} \)
41 \( 1 - 288T + 6.89e4T^{2} \)
43 \( 1 + 200iT - 7.95e4T^{2} \)
47 \( 1 - 384iT - 1.03e5T^{2} \)
53 \( 1 + 496iT - 1.48e5T^{2} \)
59 \( 1 + 128T + 2.05e5T^{2} \)
61 \( 1 + 458T + 2.26e5T^{2} \)
67 \( 1 - 496iT - 3.00e5T^{2} \)
71 \( 1 + 512T + 3.57e5T^{2} \)
73 \( 1 + 602iT - 3.89e5T^{2} \)
79 \( 1 + 1.10e3T + 4.93e5T^{2} \)
83 \( 1 + 704iT - 5.71e5T^{2} \)
89 \( 1 + 960T + 7.04e5T^{2} \)
97 \( 1 + 206iT - 9.12e5T^{2} \)
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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

−9.388295226244669679260101860064, −8.632662603453677885236345100363, −7.63544854211312086908901840672, −7.26721445541541534465783662721, −5.91377900059420246383667924249, −5.46279773327275600983828546674, −4.54749442780817415172929661341, −3.38778783659632234022091700824, −2.49993053539667706191837468908, −1.50098375176095871374592539682, 0.092160355158070527508660555315, 0.990242795512852736264149046519, 2.58044692364601156839772417517, 3.16130504692258571769513401708, 4.39437550231944096163133226722, 5.30332016137997398492004057829, 5.81005970875817812941579013912, 7.17840778314708130700105552394, 7.63083248840639964612880429709, 8.256462874117722207163324139719

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