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Properties

Label 2-525-5.4-c1-0-12
Degree $2$
Conductor $525$
Sign $-0.447 + 0.894i$
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  − 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: $-0.447 + 0.894i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
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

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

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