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Properties

Label 2-525-105.32-c1-0-3
Degree $2$
Conductor $525$
Sign $0.146 - 0.989i$
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 − 0.391i)2-s + (0.852 − 1.50i)3-s + (0.246 + 0.142i)4-s + (−1.83 + 1.86i)6-s + (−2.36 + 1.17i)7-s + (1.83 + 1.83i)8-s + (−1.54 − 2.57i)9-s + (0.791 + 0.457i)11-s + (0.425 − 0.250i)12-s + (−3.07 + 3.07i)13-s + (3.92 − 0.791i)14-s + (−2.24 − 3.88i)16-s + (0.311 + 1.16i)17-s + (1.24 + 4.35i)18-s + (−5.95 + 3.43i)19-s + ⋯
L(s)  = 1  + (−1.03 − 0.276i)2-s + (0.492 − 0.870i)3-s + (0.123 + 0.0712i)4-s + (−0.749 + 0.762i)6-s + (−0.895 + 0.444i)7-s + (0.648 + 0.648i)8-s + (−0.514 − 0.857i)9-s + (0.238 + 0.137i)11-s + (0.122 − 0.0723i)12-s + (−0.854 + 0.854i)13-s + (1.04 − 0.211i)14-s + (−0.561 − 0.971i)16-s + (0.0755 + 0.281i)17-s + (0.294 + 1.02i)18-s + (−1.36 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.146 - 0.989i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218858 + 0.188795i\)
\(L(\frac12)\) \(\approx\) \(0.218858 + 0.188795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.852 + 1.50i)T \)
5 \( 1 \)
7 \( 1 + (2.36 - 1.17i)T \)
good2 \( 1 + (1.46 + 0.391i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-0.791 - 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 + (-0.311 - 1.16i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.95 - 3.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.505 - 1.88i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.72T + 29T^{2} \)
31 \( 1 + (2.31 - 4.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.207 - 0.774i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.922iT - 41T^{2} \)
43 \( 1 + (-4.80 + 4.80i)T - 43iT^{2} \)
47 \( 1 + (-10.1 - 2.71i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (10.6 - 2.85i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.94 - 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.533 - 0.924i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.83 - 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.557iT - 71T^{2} \)
73 \( 1 + (-0.564 - 2.10i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.62 - 1.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 + (5.64 + 9.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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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

−10.84893261746583246063284176313, −9.924273129776676138701370139884, −9.155606610095040958712288694127, −8.640234842906670369670759611471, −7.62514994677515962875735763572, −6.77936575274241751266976489912, −5.79605026500005685068044419167, −4.18958088305171842183861912505, −2.65133793174310965816748443105, −1.62820758504491566505337906556, 0.21995096915658648612808004235, 2.63327639511844666324524141640, 3.86959115491870920140310004742, 4.78451529058195882012510539342, 6.27927464822723728532805250847, 7.35777548352855621560769363650, 8.138249828650951094460066542772, 9.079318832645657391505710753543, 9.592954307466629302049008985258, 10.40126387250840956830487999402

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