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Properties

Label 2-30e2-9.7-c1-0-10
Degree $2$
Conductor $900$
Sign $-0.445 + 0.895i$
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  + (−1.36 − 1.07i)3-s + (−0.0432 + 0.0748i)7-s + (0.704 + 2.91i)9-s + (0.456 − 0.791i)11-s + (−1.31 − 2.27i)13-s + 2.08·17-s + 4.93·19-s + (0.139 − 0.0555i)21-s + (−4.23 − 7.34i)23-s + (2.16 − 4.72i)27-s + (−1.19 + 2.07i)29-s + (−1.81 − 3.13i)31-s + (−1.46 + 0.587i)33-s − 5.85·37-s + (−0.649 + 4.49i)39-s + ⋯
L(s)  = 1  + (−0.785 − 0.618i)3-s + (−0.0163 + 0.0283i)7-s + (0.234 + 0.972i)9-s + (0.137 − 0.238i)11-s + (−0.364 − 0.630i)13-s + 0.506·17-s + 1.13·19-s + (0.0303 − 0.0121i)21-s + (−0.883 − 1.53i)23-s + (0.416 − 0.908i)27-s + (−0.222 + 0.385i)29-s + (−0.325 − 0.563i)31-s + (−0.255 + 0.102i)33-s − 0.962·37-s + (−0.103 + 0.720i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.445 + 0.895i$
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: \((2,\ 900,\ (\ :1/2),\ -0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468002 - 0.755173i\)
\(L(\frac12)\) \(\approx\) \(0.468002 - 0.755173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 + 1.07i)T \)
5 \( 1 \)
good7 \( 1 + (0.0432 - 0.0748i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.456 + 0.791i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.31 + 2.27i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.08T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 + (4.23 + 7.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.19 - 2.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.81 + 3.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.85T + 37T^{2} \)
41 \( 1 + (3.32 + 5.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.12 + 7.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.34 - 2.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 + (6.16 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.08 + 5.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 5.31T + 73T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.03 + 5.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.13T + 89T^{2} \)
97 \( 1 + (5.55 - 9.61i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.08115477610389477100576142202, −8.962340418169018406606463726549, −7.917454072465908906600864856449, −7.32817877813754812692894520533, −6.30916239323788522941812450411, −5.56448839559566884633536565205, −4.71737167910249996048905620846, −3.33047409404700272048681138131, −1.98040683677355152970339884005, −0.49180132686209533017322178753, 1.45372608014731236649718346941, 3.23797259287458233184581961090, 4.17927527662628418212093490259, 5.18429518830398752670545100868, 5.87483599243243128107663488354, 6.94241311002690091176158266183, 7.69421128841440810414427215549, 8.983645819920547625973594450613, 9.733326511593717473163801453423, 10.19740843993780109482048726768

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