Properties

Label 2-825-55.14-c1-0-10
Degree $2$
Conductor $825$
Sign $-0.692 - 0.721i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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  + (1.38 + 1.90i)2-s + (0.951 − 0.309i)3-s + (−1.09 + 3.37i)4-s + (1.90 + 1.38i)6-s + (0.184 + 0.0598i)7-s + (−3.47 + 1.12i)8-s + (0.809 − 0.587i)9-s + (−1.96 + 2.67i)11-s + 3.54i·12-s + (0.572 + 0.787i)13-s + (0.140 + 0.433i)14-s + (−1.21 − 0.880i)16-s + (−1.57 + 2.16i)17-s + (2.24 + 0.727i)18-s + (1.71 + 5.27i)19-s + ⋯
L(s)  = 1  + (0.979 + 1.34i)2-s + (0.549 − 0.178i)3-s + (−0.548 + 1.68i)4-s + (0.778 + 0.565i)6-s + (0.0695 + 0.0226i)7-s + (−1.22 + 0.398i)8-s + (0.269 − 0.195i)9-s + (−0.591 + 0.806i)11-s + 1.02i·12-s + (0.158 + 0.218i)13-s + (0.0376 + 0.115i)14-s + (−0.303 − 0.220i)16-s + (−0.381 + 0.525i)17-s + (0.528 + 0.171i)18-s + (0.393 + 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.692 - 0.721i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.692 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16128 + 2.72270i\)
\(L(\frac12)\) \(\approx\) \(1.16128 + 2.72270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
11 \( 1 + (1.96 - 2.67i)T \)
good2 \( 1 + (-1.38 - 1.90i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.184 - 0.0598i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.572 - 0.787i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.57 - 2.16i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.71 - 5.27i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.80iT - 23T^{2} \)
29 \( 1 + (-3.12 + 9.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.02 + 1.47i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.43 - 1.76i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.55 + 7.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + (-10.3 + 3.35i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.45 + 7.51i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.46 + 10.6i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.975 + 0.708i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.25iT - 67T^{2} \)
71 \( 1 + (4.84 + 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.14 + 1.02i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.21 + 5.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.88 - 6.72i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 7.34T + 89T^{2} \)
97 \( 1 + (-9.31 - 12.8i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.33970416906626408936497840043, −9.548881156121468651336381861069, −8.311459669286819986012221879333, −7.86321564028996318682544610111, −7.06325126430318873060950019251, −6.17924599368187610766411618545, −5.35366537243136498291601504210, −4.32320276398499548446586418560, −3.56576707437200235818769455123, −2.06948182664867249031606663320, 1.09136142090805271333771906692, 2.77321713814794721611007288589, 2.98033445554917942538480758348, 4.39588912278513227244981352667, 4.96402165255606995726419634065, 6.08682364308539708938666376053, 7.35156358742935664945870903634, 8.496851896599185107683095137547, 9.261573615812598868499007383422, 10.23317192765892593918393773762

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