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Properties

Label 2-2925-585.229-c0-0-1
Degree $2$
Conductor $2925$
Sign $0.762 + 0.646i$
Analytic cond. $1.45976$
Root an. cond. $1.20820$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (−0.866 − 0.5i)4-s − 9-s + (1.36 + 0.366i)11-s + (−0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + 17-s + (1 + i)19-s + (−0.5 + 0.866i)23-s + i·27-s + (0.5 + 0.866i)29-s + (0.366 − 1.36i)33-s + (0.866 + 0.5i)36-s + (−1 − i)37-s + ⋯
L(s)  = 1  − i·3-s + (−0.866 − 0.5i)4-s − 9-s + (1.36 + 0.366i)11-s + (−0.5 + 0.866i)12-s + (0.866 + 0.5i)13-s + (0.499 + 0.866i)16-s + 17-s + (1 + i)19-s + (−0.5 + 0.866i)23-s + i·27-s + (0.5 + 0.866i)29-s + (0.366 − 1.36i)33-s + (0.866 + 0.5i)36-s + (−1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.762 + 0.646i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (1399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129453641\)
\(L(\frac12)\) \(\approx\) \(1.129453641\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-1 - i)T + iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−8.719989455286021665875358173769, −8.270915538638889674429434392683, −7.24745123610897443832897989100, −6.64214967000613559347537010253, −5.71338460009820615678024097106, −5.31050494964648009303292952165, −3.91184080035300860464203298610, −3.46203559030012178610594380138, −1.68806623900742459623268777849, −1.21413358916088766339001458756, 0.955603273068434059797208846090, 2.93438544226094252365164458585, 3.54934751232010378990465677202, 4.22434730652825650726640144747, 5.03428052509972175811105669990, 5.79565306346995531070190823979, 6.64835563505084242887576364462, 7.81204052690610122830718116888, 8.509534481811730060147800601507, 8.977035973343150003201617384256

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