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Properties

Label 2-525-25.21-c1-0-0
Degree $2$
Conductor $525$
Sign $-0.284 + 0.958i$
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  + (−0.758 + 2.33i)2-s + (−0.809 − 0.587i)3-s + (−3.25 − 2.36i)4-s + (0.447 + 2.19i)5-s + (1.98 − 1.44i)6-s + 7-s + (4.00 − 2.91i)8-s + (0.309 + 0.951i)9-s + (−5.45 − 0.616i)10-s + (−1.26 + 3.88i)11-s + (1.24 + 3.82i)12-s + (−0.686 − 2.11i)13-s + (−0.758 + 2.33i)14-s + (0.925 − 2.03i)15-s + (1.27 + 3.91i)16-s + (−3.33 + 2.42i)17-s + ⋯
L(s)  = 1  + (−0.536 + 1.64i)2-s + (−0.467 − 0.339i)3-s + (−1.62 − 1.18i)4-s + (0.200 + 0.979i)5-s + (0.810 − 0.588i)6-s + 0.377·7-s + (1.41 − 1.02i)8-s + (0.103 + 0.317i)9-s + (−1.72 − 0.195i)10-s + (−0.380 + 1.17i)11-s + (0.358 + 1.10i)12-s + (−0.190 − 0.585i)13-s + (−0.202 + 0.623i)14-s + (0.239 − 0.525i)15-s + (0.317 + 0.977i)16-s + (−0.808 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.284 + 0.958i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233368 - 0.312600i\)
\(L(\frac12)\) \(\approx\) \(0.233368 - 0.312600i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.447 - 2.19i)T \)
7 \( 1 - T \)
good2 \( 1 + (0.758 - 2.33i)T + (-1.61 - 1.17i)T^{2} \)
11 \( 1 + (1.26 - 3.88i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.686 + 2.11i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.33 - 2.42i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (5.18 - 3.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.955 + 2.94i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.990 + 0.719i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-5.36 + 3.89i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.12 + 6.54i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.14 - 6.58i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 + (7.12 + 5.17i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.26 + 4.55i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.21 + 6.82i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.93 - 9.02i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-4.01 + 2.91i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.91 - 3.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.90 - 15.1i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.51 - 5.45i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.92 + 3.57i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (5.22 - 16.0i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.95 - 2.15i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.20173714713795555857680813624, −10.31771930373954187748801845070, −9.716458930169595593122478072088, −8.322786215861530101377510904846, −7.81956186507347127518340118110, −6.80613272324481698102443160916, −6.37557504133182146121622158543, −5.35977491914164605008899652740, −4.32482690793016977935733059387, −2.16350047204563566812595091231, 0.28563672581974136133891124791, 1.68635287333397500048875185250, 3.04398871478957756201509825466, 4.39516560922365855658454056454, 5.00217695249709954794561010212, 6.43452104446217761498582186499, 8.137025160754855277855307645147, 8.877236728064672543789311069303, 9.389625549773793077048965737504, 10.46068904747158327700396623318

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