Properties

Degree $2$
Conductor $525$
Sign $-0.447 - 0.894i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s + 4-s − 6-s + i·7-s + 3i·8-s − 9-s + 4·11-s + i·12-s − 2i·13-s − 14-s − 16-s + 6i·17-s i·18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s + 0.5·4-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s − 0.333·9-s + 1.20·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s − 0.250·16-s + 1.45i·17-s − 0.235i·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 - 0.894i$
Motivic weight: \(1\)
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898601 + 1.45396i\)
\(L(\frac12)\) \(\approx\) \(0.898601 + 1.45396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 - iT - 2T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−10.95809906583212942849790701821, −10.42812947852719999368133858680, −9.140472566501612034019629535407, −8.488328608564253859675146718757, −7.54094628244471551744421440890, −6.31475191170401825389607281693, −5.92531618569511840937896131485, −4.61125913327929135093737781904, −3.45025563682784204589111900324, −1.96998235655500263440119339869, 1.08202254048993775485563266043, 2.29669120474052286253368683505, 3.49895542701816168695234282171, 4.65287061866001724993440637193, 6.34401412539296377470059259188, 6.78234151122684636589981585666, 7.73870928607150636665463250908, 9.005568505376896954722569084493, 9.756251620087751561418383755651, 10.80239565493072587102393066548

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