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Properties

Label 2-30e2-25.14-c1-0-3
Degree $2$
Conductor $900$
Sign $-0.133 - 0.991i$
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  + (1.98 − 1.02i)5-s + 3.54i·7-s + (−1.78 + 1.29i)11-s + (−4.21 + 5.80i)13-s + (−6.05 + 1.96i)17-s + (0.715 + 2.20i)19-s + (1.27 + 1.76i)23-s + (2.90 − 4.06i)25-s + (0.262 − 0.806i)29-s + (−1.32 − 4.09i)31-s + (3.62 + 7.05i)35-s + (4.24 − 5.84i)37-s + (−1.08 − 0.790i)41-s + 8.18i·43-s + (5.75 + 1.87i)47-s + ⋯
L(s)  = 1  + (0.889 − 0.457i)5-s + 1.34i·7-s + (−0.538 + 0.390i)11-s + (−1.17 + 1.61i)13-s + (−1.46 + 0.477i)17-s + (0.164 + 0.504i)19-s + (0.266 + 0.367i)23-s + (0.581 − 0.813i)25-s + (0.0486 − 0.149i)29-s + (−0.238 − 0.734i)31-s + (0.613 + 1.19i)35-s + (0.698 − 0.961i)37-s + (−0.169 − 0.123i)41-s + 1.24i·43-s + (0.839 + 0.272i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.133 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874699 + 1.00018i\)
\(L(\frac12)\) \(\approx\) \(0.874699 + 1.00018i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.98 + 1.02i)T \)
good7 \( 1 - 3.54iT - 7T^{2} \)
11 \( 1 + (1.78 - 1.29i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.21 - 5.80i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.05 - 1.96i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.715 - 2.20i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.27 - 1.76i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.262 + 0.806i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.32 + 4.09i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.24 + 5.84i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.08 + 0.790i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.18iT - 43T^{2} \)
47 \( 1 + (-5.75 - 1.87i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.3 + 3.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.0 - 7.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.59 + 4.06i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-4.50 + 1.46i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (4.25 - 13.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.640 + 0.881i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.80 + 5.55i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-11.9 + 3.87i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.68 + 4.12i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-17.2 - 5.60i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.10927456065291520156408271078, −9.254013512218992533889198736048, −9.055007693685912124986294573448, −7.898209842436547977952034540570, −6.75936988408167523812487788723, −5.97626584639612748349932242528, −5.09195180337245204219803127865, −4.30924442473334985906214115789, −2.41158195083048251188185479001, −2.01181064896787133307935851997, 0.59048774136925598083153121338, 2.38857987851124216969751487919, 3.25261887698941872006372965612, 4.69726704692733974532731761812, 5.39352542230268801792012265257, 6.63797392187160682236891280825, 7.19979908455611590170844138820, 8.085738153305109643324477137855, 9.201208298608393633612769227268, 10.10129116735335103319600364092

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