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Properties

Label 2-30e2-9.7-c1-0-11
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: $\chi_{900} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.999 - 0.0129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93804 + 0.0125823i\)
\(L(\frac12)\) \(\approx\) \(1.93804 + 0.0125823i\)
\(L(\frac{3}{2})\) 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

−10.04966273869630461906145638928, −9.475071615684798061150364353588, −8.271442823109847933061591549287, −7.915201229668485722583066854729, −6.71891091273375517961414038778, −5.54461069880394880091943811561, −4.64220038215866381697621693073, −3.81847747402952970426362595285, −2.89955593340872857169067034619, −1.04683048544563048662701924528, 1.45418259488008027985550890913, 2.33487435911745472645248036192, 3.53484783336090373776844694161, 5.01795614093481785031073700741, 5.76802083656332146461985919909, 6.79909115707789744628792613710, 7.67376528943821992741002754432, 8.259673735134839435675789547715, 9.333395830354097040912256521598, 9.707395493705245369675255198294

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