Properties

Label 2-525-25.11-c1-0-15
Degree $2$
Conductor $525$
Sign $0.992 - 0.125i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
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 + (−0.309 + 0.951i)3-s + (0.190 − 0.587i)4-s + 2.23·5-s + (0.499 + 1.53i)6-s − 7-s + (0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s + (2.92 − 2.12i)10-s + (0.618 − 0.449i)11-s + (0.5 + 0.363i)12-s + (2.80 + 2.04i)13-s + (−1.30 + 0.951i)14-s + (−0.690 + 2.12i)15-s + (3.92 + 2.85i)16-s + (0.454 + 1.40i)17-s + ⋯
L(s)  = 1  + (0.925 − 0.672i)2-s + (−0.178 + 0.549i)3-s + (0.0954 − 0.293i)4-s + 0.999·5-s + (0.204 + 0.628i)6-s − 0.377·7-s + (0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s + (0.925 − 0.672i)10-s + (0.186 − 0.135i)11-s + (0.144 + 0.104i)12-s + (0.779 + 0.566i)13-s + (−0.349 + 0.254i)14-s + (−0.178 + 0.549i)15-s + (0.981 + 0.713i)16-s + (0.110 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47368 + 0.155631i\)
\(L(\frac12)\) \(\approx\) \(2.47368 + 0.155631i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 - 2.23T \)
7 \( 1 + T \)
good2 \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \)
11 \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.80 - 2.04i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.454 - 1.40i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.309 + 0.224i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.07 - 6.37i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.19 + 3.66i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.927 + 0.673i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.11 + 2.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-3.70 + 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.26 + 3.88i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.236 - 0.171i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.78 + 5.48i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.454 + 1.40i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.3 - 8.28i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.33 - 7.19i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4 + 12.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.61 - 2.62i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.54 + 10.9i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.93792276096069653142205904935, −10.28835532079281482478015036278, −9.192582154092166529461593558606, −8.567878420167945118961505142994, −6.91771776750852167465237574510, −5.91417708724242520864586791079, −5.12486970892762200839904860534, −4.05728632087198794572034964292, −3.12673878712068688196868780349, −1.85836653948862329741753387911, 1.36045817475820196998737946707, 3.03246709456858498475918707015, 4.37422959629813724442148635519, 5.62259359932216222501328682576, 6.02197424421362615696617875634, 6.84250419727852991433220720269, 7.84431998514043485500010396307, 9.089868481589282570386762131542, 10.00282249621521797849903070823, 10.76688449871811639144483992321

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