Properties

Label 2-2925-117.31-c0-0-0
Degree $2$
Conductor $2925$
Sign $0.546 - 0.837i$
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  − 3-s + (−0.866 − 0.5i)4-s + 9-s + (−0.366 + 1.36i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)13-s + (0.499 + 0.866i)16-s i·17-s + (−1 + i)19-s + (−0.866 − 0.5i)23-s − 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  − 3-s + (−0.866 − 0.5i)4-s + 9-s + (−0.366 + 1.36i)11-s + (0.866 + 0.5i)12-s + (0.5 − 0.866i)13-s + (0.499 + 0.866i)16-s i·17-s + (−1 + i)19-s + (−0.866 − 0.5i)23-s − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5176053404\)
\(L(\frac12)\) \(\approx\) \(0.5176053404\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
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

−9.444930611255535836425637100917, −8.016628270018677764690130909907, −7.79869212803549420742933351098, −6.42945240434180854140127473613, −6.07801557565012339760402147131, −5.08858830245255673536930640089, −4.58915883096069873916596771733, −3.84899247429965359862985175003, −2.27797494818232771576332496444, −1.04671269292018965834359191463, 0.46716688699010155285830984802, 2.00693177149277525959063363460, 3.59097158342244755556324677901, 4.03329527102991937547660542722, 4.99521539836493012706220854013, 5.75871335249062348630871336349, 6.37860111696882272943935716871, 7.30061274287630028406995192581, 8.168949176125350183546625794781, 8.834843576872041701542852124902

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