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Properties

Label 2-525-175.153-c1-0-25
Degree $2$
Conductor $525$
Sign $0.826 + 0.563i$
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  + (0.438 + 0.861i)2-s + (−0.987 + 0.156i)3-s + (0.626 − 0.862i)4-s + (−0.822 − 2.07i)5-s + (−0.568 − 0.781i)6-s + (0.329 + 2.62i)7-s + (2.92 + 0.463i)8-s + (0.951 − 0.309i)9-s + (1.42 − 1.62i)10-s + (1.12 − 3.46i)11-s + (−0.483 + 0.949i)12-s + (−6.29 − 3.20i)13-s + (−2.11 + 1.43i)14-s + (1.13 + 1.92i)15-s + (0.226 + 0.696i)16-s + (2.05 + 0.325i)17-s + ⋯
L(s)  = 1  + (0.310 + 0.608i)2-s + (−0.570 + 0.0903i)3-s + (0.313 − 0.431i)4-s + (−0.367 − 0.929i)5-s + (−0.231 − 0.319i)6-s + (0.124 + 0.992i)7-s + (1.03 + 0.163i)8-s + (0.317 − 0.103i)9-s + (0.452 − 0.512i)10-s + (0.339 − 1.04i)11-s + (−0.139 + 0.274i)12-s + (−1.74 − 0.890i)13-s + (−0.565 + 0.383i)14-s + (0.293 + 0.497i)15-s + (0.0565 + 0.174i)16-s + (0.499 + 0.0790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.826 + 0.563i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38798 - 0.428213i\)
\(L(\frac12)\) \(\approx\) \(1.38798 - 0.428213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (0.822 + 2.07i)T \)
7 \( 1 + (-0.329 - 2.62i)T \)
good2 \( 1 + (-0.438 - 0.861i)T + (-1.17 + 1.61i)T^{2} \)
11 \( 1 + (-1.12 + 3.46i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (6.29 + 3.20i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-2.05 - 0.325i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-5.52 + 4.01i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.64 + 1.85i)T + (13.5 - 18.6i)T^{2} \)
29 \( 1 + (-1.47 + 2.02i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.02 + 1.40i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.0278 + 0.0141i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-8.43 + 2.74i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.68 + 5.68i)T + 43iT^{2} \)
47 \( 1 + (-0.874 - 5.51i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.42 - 9.01i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-2.45 - 7.56i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.93 - 0.629i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.180 + 1.13i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (10.5 + 7.67i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.70 - 5.30i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (9.86 - 13.5i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.829 + 5.23i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (2.51 - 7.75i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.583 - 3.68i)T + (-92.2 + 29.9i)T^{2} \)
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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

−10.92994552570838042541914529891, −9.817933312515736096182044182068, −9.012258404952737024126423759215, −7.88207704565769284245758225118, −7.11934944206422810060144611257, −5.74705865923398488660245235179, −5.39622195553024249985847023335, −4.58472743541555683768248489861, −2.76063233570397868067277739756, −0.883427712644454921651889002460, 1.71367581853174141211525245722, 3.08778675834109747185022329444, 4.12984332568242743080011864978, 4.98868774080246327129628981756, 6.76262312166832005740418466515, 7.26087430643640642693663746401, 7.73180283469051617913171208099, 9.806036267814034907305205940501, 10.09800492066427463349581435293, 11.19190956838817852649225951557

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