Properties

Label 2-825-55.14-c1-0-8
Degree $2$
Conductor $825$
Sign $-0.750 + 0.661i$
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  + (1.62 + 2.24i)2-s + (−0.951 + 0.309i)3-s + (−1.75 + 5.40i)4-s + (−2.24 − 1.62i)6-s + (2.16 + 0.703i)7-s + (−9.71 + 3.15i)8-s + (0.809 − 0.587i)9-s + (−0.105 + 3.31i)11-s − 5.68i·12-s + (0.256 + 0.352i)13-s + (1.95 + 6.00i)14-s + (−13.7 − 9.96i)16-s + (−2.93 + 4.04i)17-s + (2.63 + 0.856i)18-s + (−1.45 − 4.46i)19-s + ⋯
L(s)  = 1  + (1.15 + 1.58i)2-s + (−0.549 + 0.178i)3-s + (−0.878 + 2.70i)4-s + (−0.915 − 0.665i)6-s + (0.818 + 0.266i)7-s + (−3.43 + 1.11i)8-s + (0.269 − 0.195i)9-s + (−0.0317 + 0.999i)11-s − 1.64i·12-s + (0.0710 + 0.0977i)13-s + (0.521 + 1.60i)14-s + (−3.42 − 2.49i)16-s + (−0.712 + 0.981i)17-s + (0.621 + 0.201i)18-s + (−0.332 − 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.750 + 0.661i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.750 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.710075 - 1.87989i\)
\(L(\frac12)\) \(\approx\) \(0.710075 - 1.87989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 \)
11 \( 1 + (0.105 - 3.31i)T \)
good2 \( 1 + (-1.62 - 2.24i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 + (-2.16 - 0.703i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.256 - 0.352i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.93 - 4.04i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.45 + 4.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.845iT - 23T^{2} \)
29 \( 1 + (-0.821 + 2.52i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.77 + 2.74i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-8.42 - 2.73i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.32 + 4.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.00iT - 43T^{2} \)
47 \( 1 + (-0.445 + 0.144i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.37 - 8.76i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.21 - 3.74i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.39 + 1.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.47iT - 67T^{2} \)
71 \( 1 + (-9.15 - 6.65i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.01 - 2.60i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.79 - 6.38i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.47 - 4.78i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
97 \( 1 + (5.08 + 6.99i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.07757370158960764278132571084, −9.654952252621884024970790997230, −8.627702144352788975052275749864, −7.962161423891413391168308798327, −7.01719438929293194091706573631, −6.34335222883391220915575077200, −5.49531612533787644669507803522, −4.50368169128087791048626702769, −4.25855094008411684226147012306, −2.52605361105195488284350067406, 0.74226785971880393844805425374, 1.92299102707153251117711773913, 3.13358513006149027102616788954, 4.19899155097277665735694553458, 4.99718387469453450259656337542, 5.76011110353382895206053028002, 6.66094269459220906465436040265, 8.178496581616421438069550461209, 9.233628806099683838614352368443, 10.20872735284847805227917857900

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