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Properties

Label 2-825-55.49-c1-0-29
Degree $2$
Conductor $825$
Sign $0.551 + 0.834i$
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  + (2.33 − 0.758i)2-s + (0.587 − 0.809i)3-s + (3.26 − 2.36i)4-s + (0.758 − 2.33i)6-s + (1.93 + 2.65i)7-s + (2.93 − 4.03i)8-s + (−0.309 − 0.951i)9-s + (2.96 + 1.47i)11-s − 4.03i·12-s + (0.297 − 0.0967i)13-s + (6.53 + 4.74i)14-s + (1.29 − 3.98i)16-s + (−4.75 − 1.54i)17-s + (−1.44 − 1.98i)18-s + (−6.03 − 4.38i)19-s + ⋯
L(s)  = 1  + (1.65 − 0.536i)2-s + (0.339 − 0.467i)3-s + (1.63 − 1.18i)4-s + (0.309 − 0.953i)6-s + (0.730 + 1.00i)7-s + (1.03 − 1.42i)8-s + (−0.103 − 0.317i)9-s + (0.894 + 0.446i)11-s − 1.16i·12-s + (0.0825 − 0.0268i)13-s + (1.74 + 1.26i)14-s + (0.323 − 0.996i)16-s + (−1.15 − 0.374i)17-s + (−0.340 − 0.468i)18-s + (−1.38 − 1.00i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.551 + 0.834i$
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: \((2,\ 825,\ (\ :1/2),\ 0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.04329 - 2.17518i\)
\(L(\frac12)\) \(\approx\) \(4.04329 - 2.17518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.96 - 1.47i)T \)
good2 \( 1 + (-2.33 + 0.758i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.93 - 2.65i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.297 + 0.0967i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.75 + 1.54i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.03 + 4.38i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.07iT - 23T^{2} \)
29 \( 1 + (4.07 - 2.96i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.06 - 3.28i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.54 - 2.13i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.77 + 6.37i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.51iT - 43T^{2} \)
47 \( 1 + (-7.05 + 9.70i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.69 - 1.52i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.41 - 5.38i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.83 + 8.73i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 15.2iT - 67T^{2} \)
71 \( 1 + (-0.949 + 2.92i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.08 - 7.00i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.67 + 5.14i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-15.4 - 5.02i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 + (-0.213 + 0.0692i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.53342348089184917493437295227, −9.049437387104486050637900166849, −8.598599511712775484568036397382, −7.03785786149323868907439339292, −6.50423404060204844714635217430, −5.40749380009593914272349138658, −4.65137917950806738742687038526, −3.70812798168553331359935160851, −2.41400067741112642164858863227, −1.85144918576773686790440130739, 1.96620552000522620315308424125, 3.43919576155880407870372551138, 4.27048973623635386066299049049, 4.58018906966083383675084988576, 6.01456760572230851529511346574, 6.52685301385791688181151620744, 7.66327563680288902760754626628, 8.340864060337010349639384838049, 9.502085689456730528672894864837, 10.82058578891718648156206709980

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