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Properties

Label 2-825-275.269-c1-0-20
Degree $2$
Conductor $825$
Sign $0.172 + 0.985i$
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  + (−1.44 − 1.99i)2-s + i·3-s + (−1.26 + 3.87i)4-s + (0.402 + 2.19i)5-s + (1.99 − 1.44i)6-s + (−1.80 − 2.48i)7-s + (4.87 − 1.58i)8-s − 9-s + (3.80 − 3.99i)10-s + (−0.359 − 3.29i)11-s + (−3.87 − 1.26i)12-s − 1.06i·13-s + (−2.33 + 7.19i)14-s + (−2.19 + 0.402i)15-s + (−3.61 − 2.62i)16-s + (−0.0124 − 0.0171i)17-s + ⋯
L(s)  = 1  + (−1.02 − 1.41i)2-s + 0.577i·3-s + (−0.630 + 1.93i)4-s + (0.180 + 0.983i)5-s + (0.814 − 0.591i)6-s + (−0.682 − 0.938i)7-s + (1.72 − 0.559i)8-s − 0.333·9-s + (1.20 − 1.26i)10-s + (−0.108 − 0.994i)11-s + (−1.11 − 0.363i)12-s − 0.296i·13-s + (−0.625 + 1.92i)14-s + (−0.567 + 0.103i)15-s + (−0.904 − 0.657i)16-s + (−0.00301 − 0.00414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.172 + 0.985i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (544, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.172 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559252 - 0.469802i\)
\(L(\frac12)\) \(\approx\) \(0.559252 - 0.469802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (-0.402 - 2.19i)T \)
11 \( 1 + (0.359 + 3.29i)T \)
good2 \( 1 + (1.44 + 1.99i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (1.80 + 2.48i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + 1.06iT - 13T^{2} \)
17 \( 1 + (0.0124 + 0.0171i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.77 - 5.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.29 + 1.07i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.97 + 6.08i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.14 - 9.68i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.47 + 1.77i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-8.18 + 5.94i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.89iT - 43T^{2} \)
47 \( 1 + 1.83iT - 47T^{2} \)
53 \( 1 + (-7.04 + 9.70i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.234 - 0.720i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 8.09T + 61T^{2} \)
67 \( 1 + (-9.27 + 12.7i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.68 - 8.26i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.164 - 0.226i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-10.9 - 7.97i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.84 - 5.29i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.52 + 10.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (1.51 - 2.08i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.11595776250399669100002644486, −9.652111992137787245466641576376, −8.627069346893624280943470492801, −7.79560617506896831823954919058, −6.78865441508251343393344190795, −5.61689460417054877425883253140, −3.79198871136574976342027167311, −3.46222860928541758511633329738, −2.42040607854857583542204681790, −0.68967337469268119941540787634, 0.955707672918570302344804393420, 2.47293798138105315553033857276, 4.63280034416432303768508339851, 5.49989851794259068239081412373, 6.23910906495199601666772435327, 7.09895318488228250076430233925, 7.80629542706952572016471240244, 8.734206119342999850284087457402, 9.413265141308264969418169555536, 9.619188199184057820770769430304

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