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Properties

Label 2-525-5.2-c2-0-21
Degree $2$
Conductor $525$
Sign $0.850 + 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.867 + 0.867i)2-s + (−1.22 + 1.22i)3-s − 2.49i·4-s − 2.12·6-s + (1.87 + 1.87i)7-s + (5.63 − 5.63i)8-s − 2.99i·9-s − 1.49·11-s + (3.05 + 3.05i)12-s + (−2.15 + 2.15i)13-s + 3.24i·14-s − 0.198·16-s + (2.96 + 2.96i)17-s + (2.60 − 2.60i)18-s − 34.8i·19-s + ⋯
L(s)  = 1  + (0.433 + 0.433i)2-s + (−0.408 + 0.408i)3-s − 0.623i·4-s − 0.354·6-s + (0.267 + 0.267i)7-s + (0.704 − 0.704i)8-s − 0.333i·9-s − 0.136·11-s + (0.254 + 0.254i)12-s + (−0.165 + 0.165i)13-s + 0.231i·14-s − 0.0124·16-s + (0.174 + 0.174i)17-s + (0.144 − 0.144i)18-s − 1.83i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.875276350\)
\(L(\frac12)\) \(\approx\) \(1.875276350\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 - 1.87i)T \)
good2 \( 1 + (-0.867 - 0.867i)T + 4iT^{2} \)
11 \( 1 + 1.49T + 121T^{2} \)
13 \( 1 + (2.15 - 2.15i)T - 169iT^{2} \)
17 \( 1 + (-2.96 - 2.96i)T + 289iT^{2} \)
19 \( 1 + 34.8iT - 361T^{2} \)
23 \( 1 + (-7.50 + 7.50i)T - 529iT^{2} \)
29 \( 1 + 37.1iT - 841T^{2} \)
31 \( 1 - 47.0T + 961T^{2} \)
37 \( 1 + (16.3 + 16.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 73.4T + 1.68e3T^{2} \)
43 \( 1 + (-0.244 + 0.244i)T - 1.84e3iT^{2} \)
47 \( 1 + (-38.9 - 38.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-33.0 + 33.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 31.6iT - 3.48e3T^{2} \)
61 \( 1 + 106.T + 3.72e3T^{2} \)
67 \( 1 + (28.6 + 28.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 15.8T + 5.04e3T^{2} \)
73 \( 1 + (-26.2 + 26.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 73.8iT - 6.24e3T^{2} \)
83 \( 1 + (58.6 - 58.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 83.2iT - 7.92e3T^{2} \)
97 \( 1 + (-103. - 103. i)T + 9.40e3iT^{2} \)
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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

−10.67215404749802202030471583242, −9.704888576536648698146601515387, −8.986746085865610804422951372525, −7.67738998147102626376545266176, −6.65147344632785026491390323852, −5.88453702774180810722261936349, −4.90207412799493623401392225583, −4.28276437425269731064375797815, −2.53711835730835382366173973702, −0.74003802532440403976268822184, 1.41055785239320435127456242504, 2.79501737324517651185458348021, 3.93826027665066678399602180974, 4.96826898346582132510837391074, 5.99707505802018641829349372961, 7.28204645081486859126686751214, 7.86220619628590446189056757797, 8.806648813862948345018967495767, 10.20902637034691249959477159259, 10.83093668966824955756555857337

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