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Properties

Label 2-525-21.17-c1-0-29
Degree $2$
Conductor $525$
Sign $0.489 + 0.871i$
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  + (1.46 − 0.846i)2-s + (−1.73 + 0.0718i)3-s + (0.433 − 0.750i)4-s + (−2.47 + 1.57i)6-s + (1.71 − 2.01i)7-s + 1.91i·8-s + (2.98 − 0.248i)9-s + (0.399 + 0.230i)11-s + (−0.695 + 1.32i)12-s − 3.38i·13-s + (0.803 − 4.40i)14-s + (2.49 + 4.31i)16-s + (2.75 − 4.76i)17-s + (4.17 − 2.89i)18-s + (3.49 − 2.01i)19-s + ⋯
L(s)  = 1  + (1.03 − 0.598i)2-s + (−0.999 + 0.0415i)3-s + (0.216 − 0.375i)4-s + (−1.01 + 0.641i)6-s + (0.647 − 0.762i)7-s + 0.678i·8-s + (0.996 − 0.0829i)9-s + (0.120 + 0.0695i)11-s + (−0.200 + 0.383i)12-s − 0.938i·13-s + (0.214 − 1.17i)14-s + (0.622 + 1.07i)16-s + (0.667 − 1.15i)17-s + (0.983 − 0.682i)18-s + (0.801 − 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.489 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66535 - 0.974919i\)
\(L(\frac12)\) \(\approx\) \(1.66535 - 0.974919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 - 0.0718i)T \)
5 \( 1 \)
7 \( 1 + (-1.71 + 2.01i)T \)
good2 \( 1 + (-1.46 + 0.846i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-0.399 - 0.230i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.38iT - 13T^{2} \)
17 \( 1 + (-2.75 + 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.49 + 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.90 + 2.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.71iT - 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.89 + 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 + 8.35T + 43T^{2} \)
47 \( 1 + (-1.57 - 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.0 - 5.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.90 - 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.458 + 0.793i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.52iT - 71T^{2} \)
73 \( 1 + (-6.75 - 3.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.58 - 6.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 + (-1.35 - 2.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.44iT - 97T^{2} \)
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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

−10.91664458738498925223113496468, −10.38189191232952985224389897501, −9.112307516504037111132306643238, −7.70642807280060767844346503535, −7.02229724582256295759756177208, −5.48858099482145937367504633156, −5.09572270887327115568647005809, −4.09881539178981795997690773092, −2.95392995764541118678987731866, −1.12468610893319335491340086108, 1.55208303187704588454345857256, 3.64558875520234680943593362868, 4.67845660205251447326777694936, 5.47046843578725803009428216583, 6.10706512432096282709110964948, 7.00985750386135979215044971056, 8.034086994889284811657967222515, 9.361471462836025391066121183793, 10.20571440443901751894802941032, 11.36405590700923385974019052413

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