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Properties

Label 2-30e2-4.3-c2-0-29
Degree $2$
Conductor $900$
Sign $-0.159 - 0.987i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 1.52i)2-s + (−0.637 − 3.94i)4-s + 0.837i·7-s + (6.83 + 4.14i)8-s + 15.7i·11-s + 5.18·13-s + (−1.27 − 1.08i)14-s + (−15.1 + 5.03i)16-s + 27.3·17-s − 17.9i·19-s + (−24.0 − 20.4i)22-s − 19.1i·23-s + (−6.72 + 7.89i)26-s + (3.30 − 0.533i)28-s − 45.6·29-s + ⋯
L(s)  = 1  + (−0.648 + 0.761i)2-s + (−0.159 − 0.987i)4-s + 0.119i·7-s + (0.854 + 0.518i)8-s + 1.43i·11-s + 0.398·13-s + (−0.0910 − 0.0775i)14-s + (−0.949 + 0.314i)16-s + 1.60·17-s − 0.945i·19-s + (−1.09 − 0.930i)22-s − 0.830i·23-s + (−0.258 + 0.303i)26-s + (0.118 − 0.0190i)28-s − 1.57·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.159 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.237593598\)
\(L(\frac12)\) \(\approx\) \(1.237593598\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 1.52i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 0.837iT - 49T^{2} \)
11 \( 1 - 15.7iT - 121T^{2} \)
13 \( 1 - 5.18T + 169T^{2} \)
17 \( 1 - 27.3T + 289T^{2} \)
19 \( 1 + 17.9iT - 361T^{2} \)
23 \( 1 + 19.1iT - 529T^{2} \)
29 \( 1 + 45.6T + 841T^{2} \)
31 \( 1 - 13.6iT - 961T^{2} \)
37 \( 1 - 15.5T + 1.36e3T^{2} \)
41 \( 1 + 13.2T + 1.68e3T^{2} \)
43 \( 1 + 27.9iT - 1.84e3T^{2} \)
47 \( 1 - 55.6iT - 2.20e3T^{2} \)
53 \( 1 - 15.5T + 2.80e3T^{2} \)
59 \( 1 - 87.6iT - 3.48e3T^{2} \)
61 \( 1 - 38T + 3.72e3T^{2} \)
67 \( 1 - 92.2iT - 4.48e3T^{2} \)
71 \( 1 - 130. iT - 5.04e3T^{2} \)
73 \( 1 - 54.7T + 5.32e3T^{2} \)
79 \( 1 - 13.6iT - 6.24e3T^{2} \)
83 \( 1 - 59.0iT - 6.88e3T^{2} \)
89 \( 1 - 39.8T + 7.92e3T^{2} \)
97 \( 1 - 168.T + 9.40e3T^{2} \)
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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

−9.960968972139234569941691566004, −9.285405597518707861167331550325, −8.454223008663907754106493201229, −7.46280266747682375529254028566, −7.01342199987857204109950093284, −5.86931128589324960637588413185, −5.09953558681711047944718936898, −4.05861445495167643182314086452, −2.38947768048867333832999635788, −1.08640403721420453585897649832, 0.61674298150837950311010147405, 1.77964926597735465389814718723, 3.35847700295071730400329389590, 3.68828718136863929734679626332, 5.32584794607575301825747540363, 6.18686228031601466873389300944, 7.58852120815499917968706996767, 8.020512259575499432654672997199, 8.966530571994993401513747674986, 9.734859872044228754669043158911

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