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Properties

Label 2-2925-13.2-c0-0-1
Degree $2$
Conductor $2925$
Sign $0.999 + 0.0257i$
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  + (−0.866 − 0.5i)4-s + (1.86 − 0.5i)7-s + i·13-s + (0.499 + 0.866i)16-s + (0.5 + 1.86i)19-s + (−1.86 − 0.5i)28-s + (1 − i)31-s + (−0.366 + 1.36i)37-s + (−1.5 − 0.866i)43-s + (2.36 − 1.36i)49-s + (0.5 − 0.866i)52-s − 0.999i·64-s + (−0.5 − 0.133i)67-s + (0.366 + 0.366i)73-s + (0.5 − 1.86i)76-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)4-s + (1.86 − 0.5i)7-s + i·13-s + (0.499 + 0.866i)16-s + (0.5 + 1.86i)19-s + (−1.86 − 0.5i)28-s + (1 − i)31-s + (−0.366 + 1.36i)37-s + (−1.5 − 0.866i)43-s + (2.36 − 1.36i)49-s + (0.5 − 0.866i)52-s − 0.999i·64-s + (−0.5 − 0.133i)67-s + (0.366 + 0.366i)73-s + (0.5 − 1.86i)76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.0257i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.999 + 0.0257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.272571823\)
\(L(\frac12)\) \(\approx\) \(1.272571823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 - iT \)
good2 \( 1 + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 + i)T - iT^{2} \)
37 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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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

−8.765443373415691193425915792053, −8.206217291841894624798259366382, −7.71418951808260056908489106388, −6.63507775389988643173489870248, −5.68750188146010335673843604056, −4.95589503814361361798896094332, −4.37180586036494635575322137992, −3.65321467985841537987425947237, −1.90333293777446260691899334665, −1.27637142743234413983658272952, 1.03343557920450659650291845988, 2.40605767381469977929799935540, 3.31231879945388330140598544339, 4.51349303883524134525798998960, 4.99995556664193919990990356420, 5.49833799782012589323576326957, 6.84355724749867222270906245856, 7.75904897425937665375648397373, 8.165791093815207825279187778509, 8.839434533857573412509744696494

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