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Properties

Label 2-825-11.5-c1-0-30
Degree $2$
Conductor $825$
Sign $-0.0571 + 0.998i$
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  + (0.0646 + 0.198i)2-s + (−0.809 − 0.587i)3-s + (1.58 − 1.14i)4-s + (0.0646 − 0.198i)6-s + (0.395 − 0.287i)7-s + (0.669 + 0.486i)8-s + (0.309 + 0.951i)9-s + (−2.66 − 1.97i)11-s − 1.95·12-s + (−1.00 − 3.10i)13-s + (0.0826 + 0.0600i)14-s + (1.15 − 3.55i)16-s + (1.02 − 3.16i)17-s + (−0.169 + 0.122i)18-s + (3.12 + 2.27i)19-s + ⋯
L(s)  = 1  + (0.0456 + 0.140i)2-s + (−0.467 − 0.339i)3-s + (0.791 − 0.574i)4-s + (0.0263 − 0.0811i)6-s + (0.149 − 0.108i)7-s + (0.236 + 0.171i)8-s + (0.103 + 0.317i)9-s + (−0.802 − 0.596i)11-s − 0.564·12-s + (−0.280 − 0.861i)13-s + (0.0220 + 0.0160i)14-s + (0.288 − 0.889i)16-s + (0.249 − 0.768i)17-s + (−0.0398 + 0.0289i)18-s + (0.718 + 0.521i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0571 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.0571 + 0.998i$
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: \((2,\ 825,\ (\ :1/2),\ -0.0571 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992309 - 1.05072i\)
\(L(\frac12)\) \(\approx\) \(0.992309 - 1.05072i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.66 + 1.97i)T \)
good2 \( 1 + (-0.0646 - 0.198i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.395 + 0.287i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.00 + 3.10i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.02 + 3.16i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.12 - 2.27i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.267T + 23T^{2} \)
29 \( 1 + (5.10 - 3.71i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.80 + 5.54i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.07 + 4.41i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.38 + 2.46i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.26T + 43T^{2} \)
47 \( 1 + (4.51 + 3.28i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.353 - 1.08i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.65 + 2.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.88 + 11.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 8.97T + 67T^{2} \)
71 \( 1 + (-1.64 + 5.05i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.81 + 5.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.466 + 1.43i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.45 - 16.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + (-5.04 - 15.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.18394355181545486730202980723, −9.342786913781812994311529721508, −7.77196100600017965661158272525, −7.62482769385505067906564571573, −6.43587416960471671890411388643, −5.55067166192279638502265994467, −5.10051701467728361682727677485, −3.33316418520368056486026136960, −2.19435199156757447958498327488, −0.72646535769137820110106271502, 1.76214240010939083707806910155, 2.93588861847041284656669364279, 4.09476260300197892639867487941, 5.06273798531266927084528907584, 6.12166575756446879719962666562, 7.06097863570276738307439590374, 7.72469003336722631219897571077, 8.747739279276925424990582814615, 9.841719774984398169801251625548, 10.46334222367394533822663693187

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