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Properties

Label 2-825-15.8-c1-0-24
Degree $2$
Conductor $825$
Sign $-0.927 + 0.374i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.70 − 1.70i)2-s + (−0.292 + 1.70i)3-s + 3.82i·4-s + (3.41 − 2.41i)6-s + (−3.41 + 3.41i)7-s + (3.12 − 3.12i)8-s + (−2.82 − i)9-s − i·11-s + (−6.53 − 1.12i)12-s + (−0.585 − 0.585i)13-s + 11.6·14-s − 2.99·16-s + (2 + 2i)17-s + (3.12 + 6.53i)18-s + 4.82i·19-s + ⋯
L(s)  = 1  + (−1.20 − 1.20i)2-s + (−0.169 + 0.985i)3-s + 1.91i·4-s + (1.39 − 0.985i)6-s + (−1.29 + 1.29i)7-s + (1.10 − 1.10i)8-s + (−0.942 − 0.333i)9-s − 0.301i·11-s + (−1.88 − 0.323i)12-s + (−0.162 − 0.162i)13-s + 3.11·14-s − 0.749·16-s + (0.485 + 0.485i)17-s + (0.735 + 1.54i)18-s + 1.10i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (518, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.292 - 1.70i)T \)
5 \( 1 \)
11 \( 1 + iT \)
good2 \( 1 + (1.70 + 1.70i)T + 2iT^{2} \)
7 \( 1 + (3.41 - 3.41i)T - 7iT^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + (-2 - 2i)T + 17iT^{2} \)
19 \( 1 - 4.82iT - 19T^{2} \)
23 \( 1 + (4.82 - 4.82i)T - 23iT^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-5.65 + 5.65i)T - 37iT^{2} \)
41 \( 1 + 0.828iT - 41T^{2} \)
43 \( 1 + (7.41 + 7.41i)T + 43iT^{2} \)
47 \( 1 + (0.828 + 0.828i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-5.41 + 5.41i)T - 67iT^{2} \)
71 \( 1 + 1.65iT - 71T^{2} \)
73 \( 1 + (7.41 + 7.41i)T + 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (4.24 - 4.24i)T - 83iT^{2} \)
89 \( 1 - 7.31T + 89T^{2} \)
97 \( 1 + (9.65 - 9.65i)T - 97iT^{2} \)
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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

−9.900019285544995577587046182031, −9.316084749052353376259938987350, −8.624853632054637841038372501569, −7.82289622495460753491461933346, −6.14860700254316074314769674939, −5.53310771678437330488790039778, −3.74578817383660621165462428931, −3.21988890744765116859114285565, −2.09194441834005791006075524131, 0, 1.06736866903146212394470344015, 2.89930760906162402174209397690, 4.59715974562372813863394394353, 6.03659367746316756295989950079, 6.54894670463704422511346021277, 7.23306289524559818203552815036, 7.75612097278575539315135710905, 8.721576776826710257419157361186, 9.689937378926328769328007599607

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