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Properties

Label 2-30e2-9.4-c1-0-9
Degree $2$
Conductor $900$
Sign $0.999 + 0.0129i$
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  + (−0.602 + 1.62i)3-s + (−1.88 − 3.26i)7-s + (−2.27 − 1.95i)9-s + (1.77 + 3.07i)11-s + (0.501 − 0.869i)13-s − 1.56·17-s + 7.21·19-s + (6.43 − 1.09i)21-s + (3.09 − 5.35i)23-s + (4.54 − 2.50i)27-s + (1.5 + 2.59i)29-s + (2.89 − 5.00i)31-s + (−6.05 + 1.02i)33-s − 0.851·37-s + (1.10 + 1.33i)39-s + ⋯
L(s)  = 1  + (−0.348 + 0.937i)3-s + (−0.712 − 1.23i)7-s + (−0.757 − 0.652i)9-s + (0.534 + 0.925i)11-s + (0.139 − 0.241i)13-s − 0.378·17-s + 1.65·19-s + (1.40 − 0.238i)21-s + (0.644 − 1.11i)23-s + (0.875 − 0.483i)27-s + (0.278 + 0.482i)29-s + (0.519 − 0.899i)31-s + (−1.05 + 0.178i)33-s − 0.139·37-s + (0.177 + 0.214i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.999 + 0.0129i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.999 + 0.0129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26169 - 0.00819134i\)
\(L(\frac12)\) \(\approx\) \(1.26169 - 0.00819134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.602 - 1.62i)T \)
5 \( 1 \)
good7 \( 1 + (1.88 + 3.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.77 - 3.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.501 + 0.869i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 - 7.21T + 19T^{2} \)
23 \( 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.89 + 5.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.851T + 37T^{2} \)
41 \( 1 + (-1.55 + 2.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.35 + 2.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.87 - 8.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (-4.83 + 8.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.10 - 7.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.45 - 2.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.21T + 71T^{2} \)
73 \( 1 + 16.7T + 73T^{2} \)
79 \( 1 + (5.49 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.09 - 8.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.33T + 89T^{2} \)
97 \( 1 + (1.93 + 3.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.07075407160122829128975146491, −9.550726888960797131259063173620, −8.658490767270672662009538977679, −7.33280793516955636580207425049, −6.76417229173035180715091771189, −5.67857726681319934283019641901, −4.58367967638056667550521701501, −3.93609693129067659156847203641, −2.90352447579408767064524730767, −0.800117601442852242742530427989, 1.10835570251460421611751717795, 2.56548603594339138642249719516, 3.44988626302783893995994983244, 5.22306578386113063699400997254, 5.82133893023944031022788818218, 6.61903439399765021993926360449, 7.43384725209940069971485952372, 8.579917254953878381627455251467, 9.044901772977502024108681784438, 10.04252999113386312118939240625

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