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Properties

Label 2-825-11.5-c1-0-2
Degree $2$
Conductor $825$
Sign $0.893 - 0.449i$
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  + (−0.564 − 1.73i)2-s + (−0.809 − 0.587i)3-s + (−1.08 + 0.786i)4-s + (−0.564 + 1.73i)6-s + (1.41 − 1.02i)7-s + (−0.978 − 0.710i)8-s + (0.309 + 0.951i)9-s + (−0.384 + 3.29i)11-s + 1.33·12-s + (1.65 + 5.09i)13-s + (−2.58 − 1.87i)14-s + (−1.50 + 4.64i)16-s + (−2.26 + 6.97i)17-s + (1.47 − 1.07i)18-s + (−4.86 − 3.53i)19-s + ⋯
L(s)  = 1  + (−0.399 − 1.22i)2-s + (−0.467 − 0.339i)3-s + (−0.541 + 0.393i)4-s + (−0.230 + 0.709i)6-s + (0.534 − 0.388i)7-s + (−0.345 − 0.251i)8-s + (0.103 + 0.317i)9-s + (−0.115 + 0.993i)11-s + 0.386·12-s + (0.459 + 1.41i)13-s + (−0.690 − 0.501i)14-s + (−0.377 + 1.16i)16-s + (−0.549 + 1.69i)17-s + (0.348 − 0.253i)18-s + (−1.11 − 0.811i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.893 - 0.449i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.893 - 0.449i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477927 + 0.113524i\)
\(L(\frac12)\) \(\approx\) \(0.477927 + 0.113524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 \)
11 \( 1 + (0.384 - 3.29i)T \)
good2 \( 1 + (0.564 + 1.73i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-1.41 + 1.02i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.65 - 5.09i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.26 - 6.97i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.86 + 3.53i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.35T + 23T^{2} \)
29 \( 1 + (4.48 - 3.25i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.00660 + 0.0203i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.907 - 0.659i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.96 - 2.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.96T + 43T^{2} \)
47 \( 1 + (-3.09 - 2.24i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.516 + 1.58i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.58 - 3.32i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.86 + 8.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.350T + 67T^{2} \)
71 \( 1 + (-1.40 + 4.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.20 + 5.95i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.792 - 2.43i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.96 + 15.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (-0.587 - 1.80i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.66482228764241946917961405496, −9.631811103661644742887558231525, −8.833499051529045038589154190347, −7.898154089105155391824565106831, −6.71801644777195787387974951792, −6.12394398955457934648617874508, −4.47296038461845617276906392136, −3.95328669914848895702221927249, −2.08705860702425948672266960918, −1.66722116742059515679923903070, 0.27932031088181049266957228683, 2.55451768952743685922020156351, 3.97681938375720365303114833736, 5.38627595000685385311298478995, 5.72247134570907226414408099777, 6.59915631279512062918900455233, 7.77684845786494362710031167238, 8.250738041477064745199962460180, 9.051806240404969220989378399354, 10.03285661455186640756573074257

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