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Properties

Label 2-525-25.6-c1-0-16
Degree $2$
Conductor $525$
Sign $0.999 - 0.0178i$
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  + (0.256 + 0.788i)2-s + (0.809 − 0.587i)3-s + (1.06 − 0.771i)4-s + (2.01 − 0.970i)5-s + (0.670 + 0.487i)6-s − 7-s + (2.22 + 1.61i)8-s + (0.309 − 0.951i)9-s + (1.28 + 1.34i)10-s + (0.559 + 1.72i)11-s + (0.405 − 1.24i)12-s + (−1.11 + 3.43i)13-s + (−0.256 − 0.788i)14-s + (1.05 − 1.96i)15-s + (0.106 − 0.328i)16-s + (−4.55 − 3.31i)17-s + ⋯
L(s)  = 1  + (0.181 + 0.557i)2-s + (0.467 − 0.339i)3-s + (0.530 − 0.385i)4-s + (0.900 − 0.433i)5-s + (0.273 + 0.199i)6-s − 0.377·7-s + (0.785 + 0.570i)8-s + (0.103 − 0.317i)9-s + (0.405 + 0.423i)10-s + (0.168 + 0.519i)11-s + (0.117 − 0.360i)12-s + (−0.309 + 0.952i)13-s + (−0.0684 − 0.210i)14-s + (0.273 − 0.508i)15-s + (0.0267 − 0.0822i)16-s + (−1.10 − 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0178i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.999 - 0.0178i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37653 + 0.0212581i\)
\(L(\frac12)\) \(\approx\) \(2.37653 + 0.0212581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-2.01 + 0.970i)T \)
7 \( 1 + T \)
good2 \( 1 + (-0.256 - 0.788i)T + (-1.61 + 1.17i)T^{2} \)
11 \( 1 + (-0.559 - 1.72i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.11 - 3.43i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.55 + 3.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-4.80 - 3.49i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.87 + 5.77i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.82 - 2.77i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.00 + 1.45i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.05 + 6.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.78 - 5.49i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.13T + 43T^{2} \)
47 \( 1 + (3.95 - 2.87i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.92 - 5.75i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.498 - 1.53i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.466 + 1.43i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-2.50 - 1.82i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-3.33 + 2.42i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.04 - 15.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.15 + 5.20i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.59 + 5.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.99 - 9.22i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.6 - 8.43i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.80066297892919467886859638558, −9.658177792401290194065464254943, −9.290780122100804943838563812483, −8.025482847733771457859426082724, −6.97657882765474004388388614342, −6.46384310519504112903751578844, −5.40952880395053003322749883821, −4.39601850926094745479912410105, −2.56008586414772367945107791579, −1.62953697224431705738440856541, 1.84023538469656442705074820683, 2.95808158586497858300108764300, 3.63884211847658445992279742940, 5.17766717644986160218993606918, 6.30020550638587780093241635090, 7.21544308559964198774796268781, 8.205883501610224940834290999538, 9.388407028515881070409424590643, 10.01663896988558078207489778843, 10.89303599273446308243381222081

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