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Properties

Label 2-825-165.134-c1-0-60
Degree $2$
Conductor $825$
Sign $-0.00132 + 0.999i$
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  + (1.11 + 0.363i)2-s + (0.451 + 1.67i)3-s + (−0.499 − 0.363i)4-s + (−0.103 + 2.03i)6-s + (−3.07 − 2.23i)7-s + (−1.80 − 2.48i)8-s + (−2.59 + 1.50i)9-s + (−0.726 − 3.23i)11-s + (0.381 − 0.999i)12-s + (−2.62 − 3.61i)14-s + (−0.736 − 2.26i)16-s + (1.38 − 0.449i)17-s + (−3.44 + 0.744i)18-s + (−3.19 − 4.39i)19-s + (2.35 − 6.15i)21-s + (0.363 − 3.88i)22-s + ⋯
L(s)  = 1  + (0.790 + 0.256i)2-s + (0.260 + 0.965i)3-s + (−0.249 − 0.181i)4-s + (−0.0421 + 0.830i)6-s + (−1.16 − 0.845i)7-s + (−0.639 − 0.880i)8-s + (−0.864 + 0.502i)9-s + (−0.219 − 0.975i)11-s + (0.110 − 0.288i)12-s + (−0.702 − 0.966i)14-s + (−0.184 − 0.566i)16-s + (0.335 − 0.108i)17-s + (−0.812 + 0.175i)18-s + (−0.732 − 1.00i)19-s + (0.513 − 1.34i)21-s + (0.0774 − 0.827i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00132 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00132 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.00132 + 0.999i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.00132 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.693356 - 0.694277i\)
\(L(\frac12)\) \(\approx\) \(0.693356 - 0.694277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.451 - 1.67i)T \)
5 \( 1 \)
11 \( 1 + (0.726 + 3.23i)T \)
good2 \( 1 + (-1.11 - 0.363i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + (3.07 + 2.23i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.38 + 0.449i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.19 + 4.39i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 9.09T + 23T^{2} \)
29 \( 1 + (7.83 + 5.69i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.54 - 4.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.11 - 7.04i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.48 - 1.80i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.35T + 43T^{2} \)
47 \( 1 + (-4.92 + 3.57i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.66 + 5.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.04 + 2.80i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.39 - 2.40i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.32iT - 67T^{2} \)
71 \( 1 + (-1.62 + 0.527i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.48 + 1.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.04 + 1.31i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.527 + 0.171i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.85iT - 89T^{2} \)
97 \( 1 + (-16.1 - 5.25i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.03133247599728829024357779586, −9.213262159420017298615305357768, −8.625267192980799478401935358072, −7.17058594862379672103068918240, −6.34601695821882698417121664082, −5.38626261532155190050577477738, −4.61029058299588183391703874465, −3.54202096908058536953624398509, −3.10062185918020751000747928675, −0.34456269570683305502888192806, 2.01886678967251626999188061085, 2.98285682677455806789953008544, 3.81422796723989251976226809437, 5.26787970619965322440631450744, 5.91620440790307823049515803012, 6.94773153899217918901819927327, 7.76332593216457411459356951926, 9.010695094684898866272416750939, 9.164127936937737205928390956381, 10.55642543087027985783380142227

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