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Properties

Label 2-30e2-45.34-c1-0-7
Degree $2$
Conductor $900$
Sign $0.917 - 0.397i$
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  + 1.73·3-s + (0.866 + 0.5i)7-s + 2.99·9-s + (−1.5 + 2.59i)11-s + (0.866 − 0.5i)13-s + 6i·17-s + 4·19-s + (1.49 + 0.866i)21-s + (2.59 − 1.5i)23-s + 5.19·27-s + (1.5 − 2.59i)29-s + (−2.5 − 4.33i)31-s + (−2.59 + 4.5i)33-s + 2i·37-s + (1.49 − 0.866i)39-s + ⋯
L(s)  = 1  + 1.00·3-s + (0.327 + 0.188i)7-s + 0.999·9-s + (−0.452 + 0.783i)11-s + (0.240 − 0.138i)13-s + 1.45i·17-s + 0.917·19-s + (0.327 + 0.188i)21-s + (0.541 − 0.312i)23-s + 1.00·27-s + (0.278 − 0.482i)29-s + (−0.449 − 0.777i)31-s + (−0.452 + 0.783i)33-s + 0.328i·37-s + (0.240 − 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31082 + 0.479282i\)
\(L(\frac12)\) \(\approx\) \(2.31082 + 0.479282i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.79 - 4.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.79 + 4.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (9.52 + 5.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.07992224583600153197905991152, −9.280268220705543416602649944008, −8.419682876613793143113610692385, −7.78780598062143123658758289477, −6.97886457688366008181130508871, −5.80431512743755988070516353672, −4.67938790766719596321986519159, −3.76506700527826423278887703269, −2.63083079928212328702409071980, −1.56281884541139202446065435860, 1.19546575042604899402041615213, 2.70877378844580335951592768834, 3.44152909619824966920828725498, 4.67537229550371026957039051836, 5.55537317746734810629500713336, 6.98807517868137866174355181792, 7.50324577513836821090992549948, 8.483492064669337425822555323328, 9.098823075597924063429267396262, 9.916510082430071664888586197412

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