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Properties

Label 2-825-55.4-c1-0-5
Degree $2$
Conductor $825$
Sign $-0.861 - 0.508i$
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  + (−0.363 + 0.5i)2-s + (−0.951 − 0.309i)3-s + (0.5 + 1.53i)4-s + (0.5 − 0.363i)6-s + (−0.726 + 0.236i)7-s + (−2.12 − 0.690i)8-s + (0.809 + 0.587i)9-s + (3.23 − 0.726i)11-s − 1.61i·12-s + (−0.726 + i)13-s + (0.145 − 0.449i)14-s + (−1.49 + 1.08i)16-s + (1.17 + 1.61i)17-s + (−0.587 + 0.190i)18-s + (−1.54 + 4.75i)19-s + ⋯
L(s)  = 1  + (−0.256 + 0.353i)2-s + (−0.549 − 0.178i)3-s + (0.250 + 0.769i)4-s + (0.204 − 0.148i)6-s + (−0.274 + 0.0892i)7-s + (−0.751 − 0.244i)8-s + (0.269 + 0.195i)9-s + (0.975 − 0.219i)11-s − 0.467i·12-s + (−0.201 + 0.277i)13-s + (0.0389 − 0.120i)14-s + (−0.374 + 0.272i)16-s + (0.285 + 0.392i)17-s + (−0.138 + 0.0450i)18-s + (−0.354 + 1.09i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.861 - 0.508i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.861 - 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205258 + 0.751612i\)
\(L(\frac12)\) \(\approx\) \(0.205258 + 0.751612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 0.726i)T \)
good2 \( 1 + (0.363 - 0.5i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + (0.726 - 0.236i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.726 - i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.17 - 1.61i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.54 - 4.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.38iT - 23T^{2} \)
29 \( 1 + (-1.80 - 5.56i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.10 - 2.30i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.781 + 2.40i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.09iT - 43T^{2} \)
47 \( 1 + (4.84 + 1.57i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.47 - 6.16i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.07 - 6.37i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.66 - 4.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.38iT - 67T^{2} \)
71 \( 1 + (-6.47 + 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-12.8 + 4.16i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.54 + 4.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.35 - 4.61i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (6.74 - 9.28i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.64755270926679503484186309407, −9.581735067099036746932015025393, −8.820127607428923130930878899390, −7.947170879495594310776090394465, −7.07112710335512842393723399492, −6.39632549226659097851219242377, −5.56414316929014889081867534518, −4.12173024466254819935781746698, −3.29779580108506983368580819429, −1.69007076768921078179694390914, 0.44216532151717620211852097535, 1.85942904767549561768513675196, 3.23586211603942235530476408559, 4.59418616235835904482479393056, 5.42308288336283585239270100851, 6.50589007380129015667436747829, 6.92445352610633359386691820787, 8.376942833768467619381023287969, 9.456072411815195752238304075702, 9.772959549899334685369524251154

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