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Properties

Label 2-525-35.9-c1-0-6
Degree $2$
Conductor $525$
Sign $0.657 + 0.753i$
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  + (−1.73 − i)2-s + (0.866 − 0.5i)3-s + (0.999 + 1.73i)4-s − 1.99·6-s + (−2.59 + 0.5i)7-s + (0.499 − 0.866i)9-s + (3 + 5.19i)11-s + (1.73 + i)12-s − 3i·13-s + (5 + 1.73i)14-s + (1.99 − 3.46i)16-s + (3.46 − 2i)17-s + (−1.73 + 0.999i)18-s + (0.5 − 0.866i)19-s + (−2 + 1.73i)21-s − 12i·22-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.499 − 0.288i)3-s + (0.499 + 0.866i)4-s − 0.816·6-s + (−0.981 + 0.188i)7-s + (0.166 − 0.288i)9-s + (0.904 + 1.56i)11-s + (0.499 + 0.288i)12-s − 0.832i·13-s + (1.33 + 0.462i)14-s + (0.499 − 0.866i)16-s + (0.840 − 0.485i)17-s + (−0.408 + 0.235i)18-s + (0.114 − 0.198i)19-s + (−0.436 + 0.377i)21-s − 2.55i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.657 + 0.753i$
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: \((2,\ 525,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787071 - 0.357946i\)
\(L(\frac12)\) \(\approx\) \(0.787071 - 0.357946i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 0.5i)T \)
good2 \( 1 + (1.73 + i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + (-3.46 + 2i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.06 - 3.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (-1.73 - i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.46 + 2i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−10.32327968832438805909886235142, −9.727235874618120320241308533800, −9.288554465884944864688396246185, −8.259391928436325381795165689721, −7.38525620926607830956039300932, −6.54135486601632489052566134698, −5.02329368734033427578974993846, −3.40106121564142083238630934460, −2.45005423925989771272156664703, −1.05841596916921040601553479351, 1.00549029580644746002583426987, 3.13302237763329420687032780019, 4.05806247224254642935848743224, 5.91821278944175291492919135726, 6.58735462151392219376968625295, 7.49603697203859800003880980761, 8.739220503478709680251628606542, 8.816782396833051452368781358044, 9.892791774504899866665188588528, 10.48870049001417470949339294361

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