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Properties

Label 2-825-55.32-c1-0-10
Degree $2$
Conductor $825$
Sign $0.312 - 0.949i$
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  + (0.366 − 0.366i)2-s + (0.707 − 0.707i)3-s + 1.73i·4-s − 0.517i·6-s + (−1 + i)7-s + (1.36 + 1.36i)8-s − 1.00i·9-s + (−1.73 + 2.82i)11-s + (1.22 + 1.22i)12-s + (3.73 + 3.73i)13-s + 0.732i·14-s − 2.46·16-s + (−3.73 + 3.73i)17-s + (−0.366 − 0.366i)18-s − 7.72·19-s + ⋯
L(s)  = 1  + (0.258 − 0.258i)2-s + (0.408 − 0.408i)3-s + 0.866i·4-s − 0.211i·6-s + (−0.377 + 0.377i)7-s + (0.482 + 0.482i)8-s − 0.333i·9-s + (−0.522 + 0.852i)11-s + (0.353 + 0.353i)12-s + (1.03 + 1.03i)13-s + 0.195i·14-s − 0.616·16-s + (−0.905 + 0.905i)17-s + (−0.0862 − 0.0862i)18-s − 1.77·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.312 - 0.949i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.312 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37838 + 0.997789i\)
\(L(\frac12)\) \(\approx\) \(1.37838 + 0.997789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
11 \( 1 + (1.73 - 2.82i)T \)
good2 \( 1 + (-0.366 + 0.366i)T - 2iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
13 \( 1 + (-3.73 - 3.73i)T + 13iT^{2} \)
17 \( 1 + (3.73 - 3.73i)T - 17iT^{2} \)
19 \( 1 + 7.72T + 19T^{2} \)
23 \( 1 + (-4.89 + 4.89i)T - 23iT^{2} \)
29 \( 1 - 7.72T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 + 2.07iT - 41T^{2} \)
43 \( 1 + (0.464 + 0.464i)T + 43iT^{2} \)
47 \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \)
53 \( 1 + (-2.07 + 2.07i)T - 53iT^{2} \)
59 \( 1 + 6.92iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 + (4.89 + 4.89i)T + 67iT^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + (-5.73 - 5.73i)T + 73iT^{2} \)
79 \( 1 + 3.58T + 79T^{2} \)
83 \( 1 + (-9.92 - 9.92i)T + 83iT^{2} \)
89 \( 1 - 1.07iT - 89T^{2} \)
97 \( 1 + (7.72 + 7.72i)T + 97iT^{2} \)
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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

−10.63825208672999860354664104456, −9.305762374262240735077855050933, −8.552734673593886972514926378682, −8.102355755764209465444933360503, −6.70318220495844930087574087774, −6.47472450853937448455114616470, −4.65234852448475201708458650805, −4.03534347644082137852655194320, −2.74406708348385364127332174194, −1.96236632257083405463541739039, 0.72512473020363872580633186111, 2.52742662745212480173979735405, 3.69472539590767992650782878391, 4.70595605584434781980903917428, 5.67052528210916180901246571316, 6.41214436667868463019899495659, 7.41189835547343731619166515747, 8.547327197052015509602541199551, 9.134499976174497213287927068998, 10.22150779392005068692739740487

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