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Properties

Label 2-825-55.14-c1-0-27
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: $\chi_{825} (124, \cdot )$
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

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

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