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Properties

Label 2-525-105.62-c0-0-1
Degree $2$
Conductor $525$
Sign $0.525 - 0.850i$
Analytic cond. $0.262009$
Root an. cond. $0.511868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + i·4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 + 0.707i)12-s + (−1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s − 1.00·36-s − 2.00i·39-s + (−0.707 − 0.707i)48-s − 1.00i·49-s + (1.41 − 1.41i)52-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + i·4-s + (0.707 − 0.707i)7-s + 1.00i·9-s + (−0.707 + 0.707i)12-s + (−1.41 − 1.41i)13-s − 16-s + 1.00·21-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s − 1.00·36-s − 2.00i·39-s + (−0.707 − 0.707i)48-s − 1.00i·49-s + (1.41 − 1.41i)52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.262009\)
Root analytic conductor: \(0.511868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099014480\)
\(L(\frac12)\) \(\approx\) \(1.099014480\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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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

−11.06108380627880486390462260870, −10.30225557912887942282927169291, −9.464701425020300512291751869324, −8.346892619081897966562496969315, −7.81686311572672741583771313411, −7.12229114070007169020647589642, −5.25884991901432943458564428020, −4.44842047778893738728465787677, −3.41988749853379855521739311917, −2.42039816419682535206851573929, 1.69822973109905766515715123762, 2.51579122388132870600032895965, 4.36040654690935255497014599943, 5.35157498113411266144568729314, 6.48035186241955739564708130023, 7.24952941496681362549494908476, 8.326894639642039016689801821975, 9.257667905558522955721180684344, 9.726748370637548568561613724359, 11.04577503457517942785670812698

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