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Properties

Label 2-525-35.9-c1-0-11
Degree $2$
Conductor $525$
Sign $-0.997 - 0.0667i$
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 + (−0.866 − 2.5i)7-s + (0.499 − 0.866i)9-s + (1 + 1.73i)11-s + (−1.73 − i)12-s − i·13-s + (−1.00 + 5.19i)14-s + (1.99 − 3.46i)16-s + (−1.73 + 0.999i)18-s + (0.5 − 0.866i)19-s + (2 + 1.73i)21-s − 3.99i·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 + (−0.327 − 0.944i)7-s + (0.166 − 0.288i)9-s + (0.301 + 0.522i)11-s + (−0.499 − 0.288i)12-s − 0.277i·13-s + (−0.267 + 1.38i)14-s + (0.499 − 0.866i)16-s + (−0.408 + 0.235i)18-s + (0.114 − 0.198i)19-s + (0.436 + 0.377i)21-s − 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.997 - 0.0667i$
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: \((2,\ 525,\ (\ :1/2),\ -0.997 - 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00768319 + 0.229952i\)
\(L(\frac12)\) \(\approx\) \(0.00768319 + 0.229952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (1.73 + i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + (-5.19 - 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.3 - 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-2.59 + 1.5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-8 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.39092639554771072308546789394, −9.656208033431892316752616866569, −9.015165641355843621649503174157, −7.77388255619842079995102857716, −7.09240027085019776790022268918, −5.80557135835649445070497932450, −4.53004630914722447925099297378, −3.32663180840104667562111604302, −1.70628688422667500909959272293, −0.22031265744412161086133938852, 1.60675948784508669872112413402, 3.42744419116655105625502276703, 5.17042692469687018896059269841, 6.15719604375254253271905355397, 6.79794039109062883203071575537, 7.79807115386811015684072114197, 8.712686364478576893402840044923, 9.270228622131828915374115653240, 10.22386193085437723987323300383, 11.12905185628392362616852199905

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