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Properties

Label 2-2925-117.70-c0-0-1
Degree $2$
Conductor $2925$
Sign $0.919 + 0.393i$
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 + (1.36 − 0.366i)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 + (1.36 − 0.366i)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 + (1.36 − 0.366i)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 + (1.36 − 0.366i)33-s + (0.866 − 0.5i)36-s + (−1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(1.45976\)
Root analytic conductor: \(1.20820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (2176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :0),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.226534046\)
\(L(\frac12)\) \(\approx\) \(2.226534046\)
\(L(1)\) 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 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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

−8.840006698026765997366492483447, −8.122893326500472633801064889397, −7.47840066465845559085162018046, −6.58232285078226761585457122929, −6.07768803864944092660103917285, −5.06048148742847462935571812467, −3.76758393252818167564240492784, −3.40350026117697719312408573591, −2.04831194871748292834444907304, −1.53253687524293825689589384525, 1.73214248961510722552873447189, 2.31581040958265386943030580885, 3.32040583190972421883903507958, 4.12208912046299003038190581477, 4.83050698297864219479368925970, 6.46231365647499356550572459331, 6.77160521404890524758369570868, 7.37482350005358557152337280336, 8.374838170751597265600774950843, 8.798269197740483286534467078343

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