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Properties

Label 2-825-275.141-c1-0-43
Degree $2$
Conductor $825$
Sign $0.895 + 0.445i$
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  + (1.30 − 0.951i)2-s + (0.809 + 0.587i)3-s + (0.190 − 0.587i)4-s + (0.690 − 2.12i)5-s + 1.61·6-s + (2.42 + 1.76i)7-s + (0.690 + 2.12i)8-s + (0.309 + 0.951i)9-s + (−1.11 − 3.44i)10-s + (−0.809 − 3.21i)11-s + (0.5 − 0.363i)12-s + (1 + 3.07i)13-s + 4.85·14-s + (1.80 − 1.31i)15-s + (3.92 + 2.85i)16-s − 0.763·17-s + ⋯
L(s)  = 1  + (0.925 − 0.672i)2-s + (0.467 + 0.339i)3-s + (0.0954 − 0.293i)4-s + (0.309 − 0.951i)5-s + 0.660·6-s + (0.917 + 0.666i)7-s + (0.244 + 0.751i)8-s + (0.103 + 0.317i)9-s + (−0.353 − 1.08i)10-s + (−0.243 − 0.969i)11-s + (0.144 − 0.104i)12-s + (0.277 + 0.853i)13-s + 1.29·14-s + (0.467 − 0.339i)15-s + (0.981 + 0.713i)16-s − 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.895 + 0.445i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.895 + 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.21333 - 0.755447i\)
\(L(\frac12)\) \(\approx\) \(3.21333 - 0.755447i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.690 + 2.12i)T \)
11 \( 1 + (0.809 + 3.21i)T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-2.42 - 1.76i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1 - 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + 0.763T + 17T^{2} \)
19 \( 1 + (0.427 + 1.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.5 + 1.08i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.64 + 5.06i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.28 - 6.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + 8.85T + 37T^{2} \)
41 \( 1 + (-0.0278 + 0.0857i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (8.16 + 5.93i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + 6.76T + 53T^{2} \)
59 \( 1 + 5.52T + 59T^{2} \)
61 \( 1 + (-2.69 + 8.28i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.927 - 2.85i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-7.59 + 5.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.64 - 8.14i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 5.38T + 83T^{2} \)
89 \( 1 + (-1.80 - 1.31i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 - 2.52T + 97T^{2} \)
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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

−10.33623351282271537052379602334, −9.183217932228079123840093101356, −8.399860986110342927797625169800, −8.153914012836457927702780749164, −6.34612848621877826433460219151, −5.23856603294510792078650407676, −4.75182229220907490860359202821, −3.81872281909296123270342472605, −2.61584728195844844788933366802, −1.65338320268291929804734447125, 1.54122079376950535838483140706, 2.95202200915168430424334925127, 4.06264824339039903348787447953, 4.94140487804094364451078787733, 5.97880275773719406422544194076, 6.81291544086306824660109300475, 7.52977671300576178227536353368, 8.126654933402361826839061024369, 9.618996570453567550606489803370, 10.30482903869181455409590523308

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