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Properties

Label 2-525-15.2-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.706 + 0.707i$
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  + (−1.78 + 1.78i)2-s + (−1.47 + 0.912i)3-s − 4.37i·4-s + (1 − 4.25i)6-s + (0.707 + 0.707i)7-s + (4.23 + 4.23i)8-s + (1.33 − 2.68i)9-s + 5.84i·11-s + (3.98 + 6.43i)12-s + (2.38 − 2.38i)13-s − 2.52·14-s − 6.37·16-s + (−3.00 + 3.00i)17-s + (2.41 + 7.17i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (−1.26 + 1.26i)2-s + (−0.850 + 0.526i)3-s − 2.18i·4-s + (0.408 − 1.73i)6-s + (0.267 + 0.267i)7-s + (1.49 + 1.49i)8-s + (0.445 − 0.895i)9-s + 1.76i·11-s + (1.15 + 1.85i)12-s + (0.661 − 0.661i)13-s − 0.674·14-s − 1.59·16-s + (−0.729 + 0.729i)17-s + (0.568 + 1.69i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.706 + 0.707i$
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: \((2,\ 525,\ (\ :1/2),\ -0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125680 - 0.302963i\)
\(L(\frac12)\) \(\approx\) \(0.125680 - 0.302963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.47 - 0.912i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.78 - 1.78i)T - 2iT^{2} \)
11 \( 1 - 5.84iT - 11T^{2} \)
13 \( 1 + (-2.38 + 2.38i)T - 13iT^{2} \)
17 \( 1 + (3.00 - 3.00i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.44 + 2.44i)T + 23iT^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (-4.76 - 4.76i)T + 37iT^{2} \)
41 \( 1 + 5.34iT - 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (-5.45 + 5.45i)T - 47iT^{2} \)
53 \( 1 + (3.77 + 3.77i)T + 53iT^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + 4.74T + 61T^{2} \)
67 \( 1 + (0.887 + 0.887i)T + 67iT^{2} \)
71 \( 1 - 8.51iT - 71T^{2} \)
73 \( 1 + (0.887 - 0.887i)T - 73iT^{2} \)
79 \( 1 - 2.11iT - 79T^{2} \)
83 \( 1 + (6.93 + 6.93i)T + 83iT^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + (2.38 + 2.38i)T + 97iT^{2} \)
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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

−10.91215994753267969823189457705, −10.24499180106323548885772030887, −9.661319559581946261662660453560, −8.695964635959169616709878217438, −7.84623907047025397049150151707, −6.86416599869808365567506831668, −6.12061994381060423422315753666, −5.25866112292262791806879974648, −4.19898478764948227180010210368, −1.60747952035980184488589513356, 0.34757772833125584440325890237, 1.53611536666868478416140773619, 2.90914643557867920977478712793, 4.24929496793713480832276954373, 5.76888810486274390897117639247, 6.90052478947517519810934128313, 7.86898921199760784602823473677, 8.728824480916934370073670138823, 9.448679530457336437921747292226, 10.72185787746949525405274518867

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