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Properties

Label 2-525-25.6-c1-0-15
Degree $2$
Conductor $525$
Sign $-0.966 + 0.254i$
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  + (−0.573 − 1.76i)2-s + (−0.809 + 0.587i)3-s + (−1.16 + 0.846i)4-s + (1.88 − 1.20i)5-s + (1.50 + 1.09i)6-s − 7-s + (−0.840 − 0.610i)8-s + (0.309 − 0.951i)9-s + (−3.20 − 2.63i)10-s + (0.745 + 2.29i)11-s + (0.445 − 1.36i)12-s + (2.10 − 6.47i)13-s + (0.573 + 1.76i)14-s + (−0.816 + 2.08i)15-s + (−1.48 + 4.57i)16-s + (4.02 + 2.92i)17-s + ⋯
L(s)  = 1  + (−0.405 − 1.24i)2-s + (−0.467 + 0.339i)3-s + (−0.582 + 0.423i)4-s + (0.842 − 0.538i)5-s + (0.612 + 0.445i)6-s − 0.377·7-s + (−0.296 − 0.215i)8-s + (0.103 − 0.317i)9-s + (−1.01 − 0.832i)10-s + (0.224 + 0.691i)11-s + (0.128 − 0.395i)12-s + (0.583 − 1.79i)13-s + (0.153 + 0.471i)14-s + (−0.210 + 0.537i)15-s + (−0.371 + 1.14i)16-s + (0.976 + 0.709i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.966 + 0.254i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.966 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121151 - 0.934942i\)
\(L(\frac12)\) \(\approx\) \(0.121151 - 0.934942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 1.20i)T \)
7 \( 1 + T \)
good2 \( 1 + (0.573 + 1.76i)T + (-1.61 + 1.17i)T^{2} \)
11 \( 1 + (-0.745 - 2.29i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.10 + 6.47i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.02 - 2.92i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.78 + 4.19i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.14 + 6.61i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.69 - 1.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.54 + 1.12i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.26 + 6.97i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.51 - 4.67i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.08T + 43T^{2} \)
47 \( 1 + (5.24 - 3.81i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.0603 + 0.0438i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.493 + 1.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.63 - 8.11i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.06 + 5.13i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-9.18 + 6.67i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.10 - 3.39i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.03 + 5.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.45 - 1.05i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.42 - 10.5i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.00 + 5.81i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.41578911887761372646035808253, −9.932818375797299131042728586835, −9.038574394695825007077312463001, −8.187542524463786598017082758945, −6.48102670466996388614159388812, −5.80282151338596118519502758433, −4.56364453819033699849486389156, −3.32705711365167950988944219910, −2.08939293876814109445552418846, −0.66609976156956555785250786956, 1.84002039877102645734327671370, 3.54122628957665935712229882451, 5.26087832722213798231474900175, 6.19273403445940015146289886667, 6.51244750311604544237595884007, 7.42693781887132765031019787210, 8.485462525618780335431698032172, 9.355211058163393447457361686528, 10.11401217637886208449648505776, 11.37442626454217087956350326399

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