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Properties

Label 2-30e2-5.4-c1-0-6
Degree $2$
Conductor $900$
Sign $-0.447 + 0.894i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  − 4i·7-s − 2i·13-s − 8·19-s − 4·31-s − 10i·37-s − 8i·43-s − 9·49-s + 14·61-s − 16i·67-s + 10i·73-s + 4·79-s − 8·91-s + 14i·97-s − 20i·103-s − 2·109-s + ⋯
L(s)  = 1  − 1.51i·7-s − 0.554i·13-s − 1.83·19-s − 0.718·31-s − 1.64i·37-s − 1.21i·43-s − 1.28·49-s + 1.79·61-s − 1.95i·67-s + 1.17i·73-s + 0.450·79-s − 0.838·91-s + 1.42i·97-s − 1.97i·103-s − 0.191·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571010 - 0.923913i\)
\(L(\frac12)\) \(\approx\) \(0.571010 - 0.923913i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−10.04029464851918290155355129228, −8.975419529647267063734345394210, −8.083068790909649331171553212913, −7.26807289819363886526191691576, −6.54957151802260648839986603555, −5.43364868088387971316187533598, −4.27205744345434838740889966837, −3.63459964505567699601710174536, −2.09726352036911341379616119879, −0.49716465820202011748401020942, 1.86995435805798286223267580506, 2.80506258313838782912658044500, 4.16912952229599195397693272089, 5.16675416129077266926274467116, 6.11075549340203458669385944203, 6.77232627370478689111305337850, 8.114318136365360461235917057505, 8.702397642281400021030677448643, 9.408910698883420418039084080263, 10.34549328126821966547238549947

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