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Properties

Label 2-825-165.38-c1-0-55
Degree $2$
Conductor $825$
Sign $0.175 + 0.984i$
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  + (0.288 − 0.566i)2-s + (0.654 − 1.60i)3-s + (0.937 + 1.29i)4-s + (−0.719 − 0.833i)6-s + (0.288 − 1.82i)7-s + (2.25 − 0.357i)8-s + (−2.14 − 2.09i)9-s + (3.31 − 0.0228i)11-s + (2.68 − 0.659i)12-s + (0.500 − 0.982i)13-s + (−0.949 − 0.689i)14-s + (−0.536 + 1.65i)16-s + (0.706 − 0.359i)17-s + (−1.80 + 0.609i)18-s + (−2.41 + 3.32i)19-s + ⋯
L(s)  = 1  + (0.204 − 0.400i)2-s + (0.377 − 0.925i)3-s + (0.468 + 0.645i)4-s + (−0.293 − 0.340i)6-s + (0.109 − 0.688i)7-s + (0.798 − 0.126i)8-s + (−0.714 − 0.699i)9-s + (0.999 − 0.00689i)11-s + (0.774 − 0.190i)12-s + (0.138 − 0.272i)13-s + (−0.253 − 0.184i)14-s + (−0.134 + 0.413i)16-s + (0.171 − 0.0872i)17-s + (−0.426 + 0.143i)18-s + (−0.554 + 0.763i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.175 + 0.984i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.175 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79941 - 1.50732i\)
\(L(\frac12)\) \(\approx\) \(1.79941 - 1.50732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.654 + 1.60i)T \)
5 \( 1 \)
11 \( 1 + (-3.31 + 0.0228i)T \)
good2 \( 1 + (-0.288 + 0.566i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-0.288 + 1.82i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.500 + 0.982i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.706 + 0.359i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.41 - 3.32i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.85 + 4.85i)T - 23iT^{2} \)
29 \( 1 + (3.25 - 2.36i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.73 + 5.34i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.12 - 7.08i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.43 + 7.48i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.67 + 5.67i)T + 43iT^{2} \)
47 \( 1 + (1.15 + 7.31i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-9.34 - 4.76i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (1.48 - 1.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.03 - 6.27i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.66 - 7.66i)T - 67iT^{2} \)
71 \( 1 + (-2.39 - 0.778i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (6.09 + 0.966i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (7.14 - 2.32i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.63 - 7.14i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 0.912T + 89T^{2} \)
97 \( 1 + (-3.52 - 1.79i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.26029386226080337894693631870, −8.930131407567205760640864966679, −8.314837229545569387348316107235, −7.27278819085966322659625540073, −6.91809270516619561264454039689, −5.83528393391125856208145748151, −4.20765656784636792673459615903, −3.47313317288288901807066981549, −2.32436042828422471694344183192, −1.17869521268814876486641170330, 1.71821542441376591312519308837, 2.98867569378076555458836144630, 4.23651466951352981636980156547, 5.12657295354343587118696085952, 5.91982537057962090198015042918, 6.83257156198362233855817364148, 7.85722765207696290177830791282, 9.077663938966852541837202507880, 9.313273230455407172771026103063, 10.41152673925055688400136065128

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