Properties

Label 2-525-7.6-c2-0-35
Degree $2$
Conductor $525$
Sign $0.879 + 0.476i$
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  + 1.71·2-s + 1.73i·3-s − 1.06·4-s + 2.96i·6-s + (3.33 − 6.15i)7-s − 8.67·8-s − 2.99·9-s + 17.0·11-s − 1.85i·12-s − 16.3i·13-s + (5.70 − 10.5i)14-s − 10.5·16-s − 13.4i·17-s − 5.13·18-s + 13.7i·19-s + ⋯
L(s)  = 1  + 0.856·2-s + 0.577i·3-s − 0.267·4-s + 0.494i·6-s + (0.476 − 0.879i)7-s − 1.08·8-s − 0.333·9-s + 1.54·11-s − 0.154i·12-s − 1.25i·13-s + (0.407 − 0.752i)14-s − 0.661·16-s − 0.789i·17-s − 0.285·18-s + 0.723i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.879 + 0.476i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.460032192\)
\(L(\frac12)\) \(\approx\) \(2.460032192\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 \)
7 \( 1 + (-3.33 + 6.15i)T \)
good2 \( 1 - 1.71T + 4T^{2} \)
11 \( 1 - 17.0T + 121T^{2} \)
13 \( 1 + 16.3iT - 169T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 13.7iT - 361T^{2} \)
23 \( 1 - 16.6T + 529T^{2} \)
29 \( 1 - 32.1T + 841T^{2} \)
31 \( 1 + 6.74iT - 961T^{2} \)
37 \( 1 - 69.2T + 1.36e3T^{2} \)
41 \( 1 + 39.7iT - 1.68e3T^{2} \)
43 \( 1 + 43.2T + 1.84e3T^{2} \)
47 \( 1 + 40.1iT - 2.20e3T^{2} \)
53 \( 1 + 22.5T + 2.80e3T^{2} \)
59 \( 1 - 81.6iT - 3.48e3T^{2} \)
61 \( 1 - 14.9iT - 3.72e3T^{2} \)
67 \( 1 + 72.0T + 4.48e3T^{2} \)
71 \( 1 + 25.7T + 5.04e3T^{2} \)
73 \( 1 + 75.0iT - 5.32e3T^{2} \)
79 \( 1 - 80.0T + 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 + 128. iT - 7.92e3T^{2} \)
97 \( 1 - 159. iT - 9.40e3T^{2} \)
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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

−10.59819436213411379490064832921, −9.745296977307815230682812482877, −8.907668452601119723908193924206, −7.932027210509926468556946203798, −6.70958308254423149629637851962, −5.67067978586713692171800421731, −4.70547797599305527502634936504, −3.97746618138244336110377196208, −3.06555733102382026195838222896, −0.859430965295749749106744866155, 1.42244131052223212232822879969, 2.83666318017686770667925182808, 4.16811646232216905343767162921, 4.93102713305944112584595484528, 6.24849300277765010336907462844, 6.59558011997336018233595230017, 8.155243992624669060522881045483, 8.995403067365263574411954411605, 9.481108250651807574240517836561, 11.27337483863411055630886876002

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