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Properties

Label 2-825-55.49-c1-0-10
Degree $2$
Conductor $825$
Sign $0.872 - 0.489i$
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  + (−0.587 + 0.190i)2-s + (0.587 − 0.809i)3-s + (−1.30 + 0.951i)4-s + (−0.190 + 0.587i)6-s + (1.76 + 2.42i)7-s + (1.31 − 1.80i)8-s + (−0.309 − 0.951i)9-s + (1.69 − 2.85i)11-s + 1.61i·12-s + (−1.67 + 0.545i)13-s + (−1.5 − 1.08i)14-s + (0.572 − 1.76i)16-s + (−1.53 − 0.5i)17-s + (0.363 + 0.5i)18-s + (4.73 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.135i)2-s + (0.339 − 0.467i)3-s + (−0.654 + 0.475i)4-s + (−0.0779 + 0.239i)6-s + (0.666 + 0.917i)7-s + (0.464 − 0.639i)8-s + (−0.103 − 0.317i)9-s + (0.509 − 0.860i)11-s + 0.467i·12-s + (−0.465 + 0.151i)13-s + (−0.400 − 0.291i)14-s + (0.143 − 0.440i)16-s + (−0.373 − 0.121i)17-s + (0.0856 + 0.117i)18-s + (1.08 + 0.789i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.872 - 0.489i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.872 - 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25113 + 0.326938i\)
\(L(\frac12)\) \(\approx\) \(1.25113 + 0.326938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
11 \( 1 + (-1.69 + 2.85i)T \)
good2 \( 1 + (0.587 - 0.190i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.76 - 2.42i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.67 - 0.545i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.53 + 0.5i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.73 - 3.44i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 3.47iT - 23T^{2} \)
29 \( 1 + (-3.61 + 2.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.881 - 2.71i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.138 - 0.190i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.66 - 7.02i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.23iT - 43T^{2} \)
47 \( 1 + (0.951 - 1.30i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.14 + 2.97i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.35 + 6.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.42 + 7.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 9.56iT - 67T^{2} \)
71 \( 1 + (1.71 - 5.29i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.90 - 2.61i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.92 + 9.00i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.673 - 0.218i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.527T + 89T^{2} \)
97 \( 1 + (13.3 - 4.33i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.911147530419775627587681655717, −9.299121319582635820185316344796, −8.466371673679721596453451314728, −8.018359898569403357876183725571, −7.10203615070833533092102575646, −5.95073494694496121378002988225, −4.98017394013264737195658428392, −3.80200570354282909564091179989, −2.70076932537563323897860230178, −1.17872831476460790667965119267, 0.922557076904206370679947408226, 2.34638780384752875802484983812, 4.03664890969670483880164452448, 4.60425772608453408754720450499, 5.46750633988209657620780599131, 6.98488422986225912150042186529, 7.68127696626706058268937263069, 8.714147976586857982627026732927, 9.322434433794114789758188953752, 10.16905597248941998795170576567

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