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Properties

Label 2-825-165.74-c1-0-64
Degree $2$
Conductor $825$
Sign $0.211 - 0.977i$
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  + (−1.11 − 1.53i)2-s + (0.0877 − 1.72i)3-s + (−0.499 + 1.53i)4-s + (−2.76 + 1.79i)6-s + (0.726 − 2.23i)7-s + (−0.690 + 0.224i)8-s + (−2.98 − 0.303i)9-s + (−3.07 − 1.23i)11-s + (2.61 + 0.999i)12-s + (−4.25 + 1.38i)14-s + (3.73 + 2.71i)16-s + (3.61 − 4.97i)17-s + (2.86 + 4.93i)18-s + (−4.30 + 1.40i)19-s + (−3.80 − 1.45i)21-s + (1.53 + 6.11i)22-s + ⋯
L(s)  = 1  + (−0.790 − 1.08i)2-s + (0.0506 − 0.998i)3-s + (−0.249 + 0.769i)4-s + (−1.12 + 0.734i)6-s + (0.274 − 0.845i)7-s + (−0.244 + 0.0793i)8-s + (−0.994 − 0.101i)9-s + (−0.927 − 0.372i)11-s + (0.755 + 0.288i)12-s + (−1.13 + 0.369i)14-s + (0.934 + 0.678i)16-s + (0.877 − 1.20i)17-s + (0.676 + 1.16i)18-s + (−0.988 + 0.321i)19-s + (−0.830 − 0.317i)21-s + (0.328 + 1.30i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324115 + 0.261352i\)
\(L(\frac12)\) \(\approx\) \(0.324115 + 0.261352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0877 + 1.72i)T \)
5 \( 1 \)
11 \( 1 + (3.07 + 1.23i)T \)
good2 \( 1 + (1.11 + 1.53i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (-0.726 + 2.23i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.61 + 4.97i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.30 - 1.40i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.09T + 23T^{2} \)
29 \( 1 + (2.21 - 6.80i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.04 + 2.93i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.47 + 1.45i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.224 + 0.690i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.33T + 43T^{2} \)
47 \( 1 + (-1.57 - 4.84i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.16 - 4.47i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.20 - 1.69i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.89 + 6.74i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + (6.88 - 9.47i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.224 - 0.690i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.54 - 2.12i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.47 + 13.0i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.854iT - 89T^{2} \)
97 \( 1 + (-9.99 - 13.7i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.716431579009749035064873054307, −8.705777650416723318472552601284, −7.985901670663326224546550925416, −7.30665771079849533806419083766, −6.16641667854170600941040564118, −5.11672680161962732806930899845, −3.48945834683231543755542359130, −2.56030636323506146672105055590, −1.42805961569699641791150076930, −0.26421825354052929161705867493, 2.38033932452367962127719950642, 3.68402649236685877814467378950, 4.99304501828399205445259690173, 5.72069495528597724484598375883, 6.50917900134598619203405577256, 7.84173218053102934136805501811, 8.358488812997659240360036307301, 8.937961317187334956733060359905, 10.06929717564646471125799743144, 10.28982799343064497595579416350

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