Properties

Label 2-825-15.8-c1-0-38
Degree $2$
Conductor $825$
Sign $0.229 - 0.973i$
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  + (1.22 + 1.22i)2-s + (1.22 + 1.22i)3-s + 0.999i·4-s + 2.99i·6-s + (2.44 − 2.44i)7-s + (1.22 − 1.22i)8-s + 2.99i·9-s + i·11-s + (−1.22 + 1.22i)12-s + (−2.44 − 2.44i)13-s + 5.99·14-s + 5·16-s + (4.89 + 4.89i)17-s + (−3.67 + 3.67i)18-s − 2i·19-s + ⋯
L(s)  = 1  + (0.866 + 0.866i)2-s + (0.707 + 0.707i)3-s + 0.499i·4-s + 1.22i·6-s + (0.925 − 0.925i)7-s + (0.433 − 0.433i)8-s + 0.999i·9-s + 0.301i·11-s + (−0.353 + 0.353i)12-s + (−0.679 − 0.679i)13-s + 1.60·14-s + 1.25·16-s + (1.18 + 1.18i)17-s + (−0.866 + 0.866i)18-s − 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.229 - 0.973i$
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: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.66413 + 2.10844i\)
\(L(\frac12)\) \(\approx\) \(2.66413 + 2.10844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 \)
11 \( 1 - iT \)
good2 \( 1 + (-1.22 - 1.22i)T + 2iT^{2} \)
7 \( 1 + (-2.44 + 2.44i)T - 7iT^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 + (-4.89 - 4.89i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (4.89 - 4.89i)T - 23iT^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (7.34 + 7.34i)T + 43iT^{2} \)
47 \( 1 + (4.89 + 4.89i)T + 47iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (-2.44 + 2.44i)T - 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (7.34 - 7.34i)T - 83iT^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (9.79 - 9.79i)T - 97iT^{2} \)
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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

−10.18738359232304750166572362043, −9.825494889937924797867314397816, −8.330483421693273147114096584837, −7.73509867036196611762681457297, −7.14396192436113412820275691100, −5.69106198843604375285769750219, −5.07780873691497002046649122971, −4.14518850466171904284743629190, −3.49608128935490271051702355809, −1.71341675491868712290049234662, 1.58593089089784044143061014768, 2.44754546425585131364813554200, 3.28767433457833897053219200939, 4.51072760007118927747464899286, 5.35249808137406888784794079140, 6.46491062830265175332087060097, 7.85891486738345580524753836437, 8.064662033249326447640412470262, 9.256891828434094937155750376779, 10.05798392665009399755914169287

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