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Properties

Label 2-1800-200.37-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.827 + 0.562i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (0.156 + 0.987i)5-s + (−1.26 − 1.26i)7-s + (0.891 − 0.453i)8-s + (0.309 + 0.951i)10-s + (1.16 − 1.59i)11-s + (−1.44 − 1.04i)14-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)20-s + (0.896 − 1.76i)22-s + (−0.951 + 0.309i)25-s + (−1.58 − 0.809i)28-s + (0.437 + 1.34i)29-s + (−0.587 + 1.80i)31-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (0.156 + 0.987i)5-s + (−1.26 − 1.26i)7-s + (0.891 − 0.453i)8-s + (0.309 + 0.951i)10-s + (1.16 − 1.59i)11-s + (−1.44 − 1.04i)14-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)20-s + (0.896 − 1.76i)22-s + (−0.951 + 0.309i)25-s + (−1.58 − 0.809i)28-s + (0.437 + 1.34i)29-s + (−0.587 + 1.80i)31-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.827 + 0.562i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.827 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.040722437\)
\(L(\frac12)\) \(\approx\) \(2.040722437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 + 0.156i)T \)
3 \( 1 \)
5 \( 1 + (-0.156 - 0.987i)T \)
good7 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
11 \( 1 + (-1.16 + 1.59i)T + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.550 + 0.280i)T + (0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)T^{2} \)
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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

−9.638935967367129521031434268299, −8.643645633122293659701677026578, −7.31722375379722224009977606962, −6.75905058679485614292429353346, −6.35797071278338639551206506530, −5.46336435155787424586311140702, −4.08757700973510664690000723265, −3.39606869110169052523420321160, −3.03939895569984911033418252825, −1.28256989055274864656887971436, 1.82503051765450891098729033790, 2.65005575802921222235232569941, 3.96892197884780457403399458638, 4.51095668271484310525002847950, 5.63082471685750181578129078185, 6.10601852937052284052218404371, 6.91331008421149839833974544952, 7.83752928366725268925100048692, 8.897227820879674167235534799861, 9.529644692907562302252056924808

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