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Properties

Label 2-525-105.104-c1-0-37
Degree $2$
Conductor $525$
Sign $0.968 + 0.247i$
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  + 1.73·2-s + 1.73·3-s + 0.999·4-s + 2.99·6-s + (1.73 − 2i)7-s − 1.73·8-s + 2.99·9-s − 3.46i·11-s + 1.73·12-s + (2.99 − 3.46i)14-s − 5·16-s + 6i·17-s + 5.19·18-s + 3.46i·19-s + (2.99 − 3.46i)21-s − 5.99i·22-s + ⋯
L(s)  = 1  + 1.22·2-s + 1.00·3-s + 0.499·4-s + 1.22·6-s + (0.654 − 0.755i)7-s − 0.612·8-s + 0.999·9-s − 1.04i·11-s + 0.500·12-s + (0.801 − 0.925i)14-s − 1.25·16-s + 1.45i·17-s + 1.22·18-s + 0.794i·19-s + (0.654 − 0.755i)21-s − 1.27i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.968 + 0.247i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.44620 - 0.433168i\)
\(L(\frac12)\) \(\approx\) \(3.44620 - 0.433168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−10.81304738211801449592937103315, −10.13098799940318061059130328801, −8.703827440457794117269020499620, −8.311075628733440204820223648835, −7.11259579238074505442502198256, −6.04741784685578589167522285886, −4.95151852031948174821800403142, −3.84294620227438735573679925753, −3.39952301725385244109906433575, −1.75446713377922185508226898441, 2.16764245529004211572156235362, 2.97719156167324051890750767877, 4.40061393218100143317930118227, 4.84012215124110855288502951239, 6.11191443730594698930135763594, 7.26680344576942928881004047252, 8.138354832461346821127631591338, 9.279806900531329628628486152042, 9.669582419526836412849286119393, 11.25637303690635835645898356793

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