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Properties

Label 2-825-55.49-c1-0-4
Degree $2$
Conductor $825$
Sign $-0.448 - 0.893i$
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  + (1.40 − 0.456i)2-s + (−0.587 + 0.809i)3-s + (0.147 − 0.107i)4-s + (−0.456 + 1.40i)6-s + (1.34 + 1.85i)7-s + (−1.57 + 2.17i)8-s + (−0.309 − 0.951i)9-s + (−3.12 + 1.12i)11-s + 0.182i·12-s + (−2.03 + 0.661i)13-s + (2.74 + 1.99i)14-s + (−1.33 + 4.11i)16-s + (0.517 + 0.168i)17-s + (−0.868 − 1.19i)18-s + (−1.76 − 1.28i)19-s + ⋯
L(s)  = 1  + (0.993 − 0.322i)2-s + (−0.339 + 0.467i)3-s + (0.0737 − 0.0535i)4-s + (−0.186 + 0.573i)6-s + (0.509 + 0.701i)7-s + (−0.558 + 0.768i)8-s + (−0.103 − 0.317i)9-s + (−0.940 + 0.339i)11-s + 0.0526i·12-s + (−0.564 + 0.183i)13-s + (0.733 + 0.532i)14-s + (−0.334 + 1.02i)16-s + (0.125 + 0.0408i)17-s + (−0.204 − 0.281i)18-s + (−0.405 − 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.448 - 0.893i$
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: \((2,\ 825,\ (\ :1/2),\ -0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761520 + 1.23374i\)
\(L(\frac12)\) \(\approx\) \(0.761520 + 1.23374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.12 - 1.12i)T \)
good2 \( 1 + (-1.40 + 0.456i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.34 - 1.85i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.03 - 0.661i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.517 - 0.168i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.76 + 1.28i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2.03iT - 23T^{2} \)
29 \( 1 + (8.04 - 5.84i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.09 - 6.44i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.18 - 7.13i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.47 - 1.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.620iT - 43T^{2} \)
47 \( 1 + (0.222 - 0.305i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-11.0 + 3.58i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (6.53 - 4.74i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.69 + 8.29i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.75iT - 67T^{2} \)
71 \( 1 + (4.63 - 14.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.61 - 6.35i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.85 - 8.77i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-8.98 - 2.92i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.583T + 89T^{2} \)
97 \( 1 + (-5.11 + 1.66i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.75628008279159599949158525748, −9.761461333100178497630191783572, −8.811559053684097170937266784205, −8.086641399235598354275538226437, −6.85106867787146543495215441838, −5.61396503282232246466260170546, −5.08206522862766146482470220586, −4.37418112820228684874191490663, −3.15110260791806422020383290457, −2.18338743914997963893068410056, 0.51653967734795625736090700280, 2.35402137672180916601801105866, 3.78806197509656536914656607701, 4.63388021064360618021831698656, 5.59066193620609980190476546849, 6.12142472483180885348687293276, 7.48248799333193180653378948610, 7.71569143372856594924034309610, 9.137593785324221234107564816189, 10.09669502613082552743463281060

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