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Properties

Label 2-525-35.9-c1-0-22
Degree $2$
Conductor $525$
Sign $-0.251 + 0.967i$
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  + (0.427 + 0.247i)2-s + (0.866 − 0.5i)3-s + (−0.877 − 1.52i)4-s + 0.494·6-s + (−1.86 + 1.87i)7-s − 1.85i·8-s + (0.499 − 0.866i)9-s + (−2.66 − 4.62i)11-s + (−1.52 − 0.877i)12-s − 5.09i·13-s + (−1.26 + 0.343i)14-s + (−1.29 + 2.24i)16-s + (0.303 − 0.175i)17-s + (0.427 − 0.247i)18-s + (1.38 − 2.39i)19-s + ⋯
L(s)  = 1  + (0.302 + 0.174i)2-s + (0.499 − 0.288i)3-s + (−0.438 − 0.760i)4-s + 0.201·6-s + (−0.704 + 0.709i)7-s − 0.656i·8-s + (0.166 − 0.288i)9-s + (−0.804 − 1.39i)11-s + (−0.438 − 0.253i)12-s − 1.41i·13-s + (−0.337 + 0.0917i)14-s + (−0.324 + 0.561i)16-s + (0.0735 − 0.0424i)17-s + (0.100 − 0.0582i)18-s + (0.317 − 0.549i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824864 - 1.06699i\)
\(L(\frac12)\) \(\approx\) \(0.824864 - 1.06699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (1.86 - 1.87i)T \)
good2 \( 1 + (-0.427 - 0.247i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (2.66 + 4.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (-0.303 + 0.175i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.38 + 2.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.50 - 3.75i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.05 + 5.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.05 - 1.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.86T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 + (-7.66 - 4.42i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.47 + 3.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.42 - 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.25 + 1.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (6.64 - 3.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.18 + 2.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.87iT - 83T^{2} \)
89 \( 1 + (2.17 - 3.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.49iT - 97T^{2} \)
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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

−10.53754908325618844879174283172, −9.587521336966158565279297375426, −8.891473593937752549044959872886, −8.003690978606615883248818229190, −6.85705857006448729484443061484, −5.60344730085161454157012267515, −5.42560153743368355243111004274, −3.58713002166623320756983829879, −2.72115410774127045642111610707, −0.67954874727708090024960517992, 2.21423578669554794080523883410, 3.44180104094434969359843925900, 4.28955375269860590647067902172, 5.11276445263478367967627062320, 6.93258479680184833834288258591, 7.38257015918304456982678277204, 8.560994511697630336396899774316, 9.374623706713533829330948924924, 10.09770642848254583800836875318, 11.08770458464779066195885451008

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