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Properties

Label 2-525-15.2-c1-0-4
Degree $2$
Conductor $525$
Sign $-0.961 + 0.275i$
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  + (−0.723 + 0.723i)2-s + (0.994 + 1.41i)3-s + 0.951i·4-s + (−1.74 − 0.307i)6-s + (−0.707 − 0.707i)7-s + (−2.13 − 2.13i)8-s + (−1.02 + 2.81i)9-s − 3.63i·11-s + (−1.34 + 0.946i)12-s + (−4.19 + 4.19i)13-s + 1.02·14-s + 1.19·16-s + (−4.61 + 4.61i)17-s + (−1.30 − 2.78i)18-s + 3.77i·19-s + ⋯
L(s)  = 1  + (−0.511 + 0.511i)2-s + (0.573 + 0.818i)3-s + 0.475i·4-s + (−0.713 − 0.125i)6-s + (−0.267 − 0.267i)7-s + (−0.755 − 0.755i)8-s + (−0.341 + 0.939i)9-s − 1.09i·11-s + (−0.389 + 0.273i)12-s + (−1.16 + 1.16i)13-s + 0.273·14-s + 0.297·16-s + (−1.11 + 1.11i)17-s + (−0.306 − 0.655i)18-s + 0.865i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.961 + 0.275i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106176 - 0.756850i\)
\(L(\frac12)\) \(\approx\) \(0.106176 - 0.756850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.994 - 1.41i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.723 - 0.723i)T - 2iT^{2} \)
11 \( 1 + 3.63iT - 11T^{2} \)
13 \( 1 + (4.19 - 4.19i)T - 13iT^{2} \)
17 \( 1 + (4.61 - 4.61i)T - 17iT^{2} \)
19 \( 1 - 3.77iT - 19T^{2} \)
23 \( 1 + (1.81 + 1.81i)T + 23iT^{2} \)
29 \( 1 + 1.13T + 29T^{2} \)
31 \( 1 - 3.62T + 31T^{2} \)
37 \( 1 + (-7.24 - 7.24i)T + 37iT^{2} \)
41 \( 1 - 0.314iT - 41T^{2} \)
43 \( 1 + (1.06 - 1.06i)T - 43iT^{2} \)
47 \( 1 + (-4.48 + 4.48i)T - 47iT^{2} \)
53 \( 1 + (-1.44 - 1.44i)T + 53iT^{2} \)
59 \( 1 - 8.70T + 59T^{2} \)
61 \( 1 - 3.08T + 61T^{2} \)
67 \( 1 + (9.67 + 9.67i)T + 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-0.710 + 0.710i)T - 73iT^{2} \)
79 \( 1 - 7.30iT - 79T^{2} \)
83 \( 1 + (-9.58 - 9.58i)T + 83iT^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 + (-5.28 - 5.28i)T + 97iT^{2} \)
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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

−11.19258187564441783429662261823, −10.14305982114177250263650234915, −9.455496255163666280040688819241, −8.560289501197012042175962299104, −8.086889491477766552738610775982, −6.93242028573843492039530609029, −6.02784237387573319180598512049, −4.45356821373515288431946177560, −3.71180925849070752334206908758, −2.47203506906140153572548254255, 0.46088615575313144659371229606, 2.20414169581562998167477199655, 2.76853063170811448877599340522, 4.69388555207510744768495352756, 5.80446318013083587736435109487, 6.97233597836344252994078972780, 7.65085479170804408479892042649, 8.862190419383146838800185908520, 9.481386623922611414559017027977, 10.14512583224599121000201661520

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