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Properties

Label 2-525-21.20-c1-0-18
Degree $2$
Conductor $525$
Sign $0.586 - 0.810i$
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  + 2.52i·2-s + (−1.68 + 0.396i)3-s − 4.37·4-s + (−1 − 4.25i)6-s + (2 − 1.73i)7-s − 5.98i·8-s + (2.68 − 1.33i)9-s − 0.792i·11-s + (7.37 − 1.73i)12-s − 5.84i·13-s + (4.37 + 5.04i)14-s + 6.37·16-s + 1.37·17-s + (3.37 + 6.78i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  + 1.78i·2-s + (−0.973 + 0.228i)3-s − 2.18·4-s + (−0.408 − 1.73i)6-s + (0.755 − 0.654i)7-s − 2.11i·8-s + (0.895 − 0.445i)9-s − 0.238i·11-s + (2.12 − 0.500i)12-s − 1.61i·13-s + (1.16 + 1.34i)14-s + 1.59·16-s + 0.332·17-s + (0.794 + 1.59i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.586 - 0.810i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.586 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760349 + 0.388376i\)
\(L(\frac12)\) \(\approx\) \(0.760349 + 0.388376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 - 0.396i)T \)
5 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 - 2.52iT - 2T^{2} \)
11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 + 5.84iT - 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 + 4.25iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 4.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6.74T + 43T^{2} \)
47 \( 1 - 7.37T + 47T^{2} \)
53 \( 1 + 8.51iT - 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 3.37T + 79T^{2} \)
83 \( 1 + 5.48T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 - 1.08iT - 97T^{2} \)
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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

−10.76717604032467851209623303008, −10.10101660470867507239684575361, −8.939057509243969610035356064897, −7.924558124787805982572716785283, −7.33661346892967501029086713857, −6.39149014839391517138188717269, −5.41633828393702255261623700285, −4.95523922174284795939583895398, −3.77914869807136625027393278089, −0.64796113728359139994112677424, 1.40143745429362552810299998003, 2.25554641029650655652788535509, 3.96551574810931030896192406226, 4.75464152156747080817414625846, 5.73381317262294266287031793152, 7.07909835709139186926625826554, 8.420229842639281222103523356802, 9.311117927282313668064670165013, 10.15429961492002573184339312793, 11.00862629020499991342539162168

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