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Properties

Label 2-525-21.17-c1-0-31
Degree $2$
Conductor $525$
Sign $0.977 + 0.211i$
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  + (0.780 − 0.450i)2-s + (1.71 − 0.248i)3-s + (−0.593 + 1.02i)4-s + (1.22 − 0.966i)6-s + (−0.105 − 2.64i)7-s + 2.87i·8-s + (2.87 − 0.853i)9-s + (5.46 + 3.15i)11-s + (−0.761 + 1.91i)12-s − 3.77i·13-s + (−1.27 − 2.01i)14-s + (0.107 + 0.185i)16-s + (−1.88 + 3.27i)17-s + (1.86 − 1.96i)18-s + (−4.57 + 2.64i)19-s + ⋯
L(s)  = 1  + (0.551 − 0.318i)2-s + (0.989 − 0.143i)3-s + (−0.296 + 0.514i)4-s + (0.500 − 0.394i)6-s + (−0.0398 − 0.999i)7-s + 1.01i·8-s + (0.958 − 0.284i)9-s + (1.64 + 0.950i)11-s + (−0.219 + 0.551i)12-s − 1.04i·13-s + (−0.340 − 0.538i)14-s + (0.0268 + 0.0464i)16-s + (−0.458 + 0.793i)17-s + (0.438 − 0.462i)18-s + (−1.05 + 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.977 + 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53974 - 0.271247i\)
\(L(\frac12)\) \(\approx\) \(2.53974 - 0.271247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.248i)T \)
5 \( 1 \)
7 \( 1 + (0.105 + 2.64i)T \)
good2 \( 1 + (-0.780 + 0.450i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-5.46 - 3.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.77iT - 13T^{2} \)
17 \( 1 + (1.88 - 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.57 - 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.87 + 3.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.06iT - 29T^{2} \)
31 \( 1 + (-0.349 - 0.201i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.668 - 1.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 + 4.84T + 43T^{2} \)
47 \( 1 + (-0.970 - 1.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.24 + 0.720i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.60 - 2.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.55 + 2.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.21iT - 71T^{2} \)
73 \( 1 + (2.18 + 1.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.05 + 5.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (0.590 + 1.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.10iT - 97T^{2} \)
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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

−10.78036021803173240965322407995, −9.959910521328924860277494537879, −8.902402065874161387261486884587, −8.241080511323980843778736684088, −7.29157289722959371870200039365, −6.43152323217214283950138710555, −4.55490025671610156563508700599, −4.05216902032053663674431084092, −3.10904273422636299487132481705, −1.66436517452125308140821011554, 1.61860029609932339897986388949, 3.16304777225552236835899164785, 4.20997327907683585296974251028, 5.11717416679933349154804556951, 6.43932213116886592202853335515, 6.91026394888702307116509340719, 8.629354349116780603879751325999, 9.084407187344075104718857829782, 9.542424516516046870714757269499, 10.96196814923635786910698614887

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