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Properties

Label 2-525-175.52-c1-0-27
Degree $2$
Conductor $525$
Sign $-0.192 + 0.981i$
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  + (−0.0754 − 0.196i)2-s + (−0.0523 + 0.998i)3-s + (1.45 − 1.30i)4-s + (−2.18 + 0.469i)5-s + (0.200 − 0.0650i)6-s + (−2.39 + 1.12i)7-s + (−0.742 − 0.378i)8-s + (−0.994 − 0.104i)9-s + (0.257 + 0.394i)10-s + (−0.565 − 5.38i)11-s + (1.23 + 1.51i)12-s + (−0.378 − 2.38i)13-s + (0.402 + 0.385i)14-s + (−0.354 − 2.20i)15-s + (0.390 − 3.71i)16-s + (1.31 − 0.852i)17-s + ⋯
L(s)  = 1  + (−0.0533 − 0.139i)2-s + (−0.0302 + 0.576i)3-s + (0.726 − 0.654i)4-s + (−0.977 + 0.209i)5-s + (0.0817 − 0.0265i)6-s + (−0.904 + 0.426i)7-s + (−0.262 − 0.133i)8-s + (−0.331 − 0.0348i)9-s + (0.0813 + 0.124i)10-s + (−0.170 − 1.62i)11-s + (0.355 + 0.438i)12-s + (−0.104 − 0.662i)13-s + (0.107 + 0.103i)14-s + (−0.0914 − 0.570i)15-s + (0.0976 − 0.928i)16-s + (0.318 − 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559895 - 0.680075i\)
\(L(\frac12)\) \(\approx\) \(0.559895 - 0.680075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0523 - 0.998i)T \)
5 \( 1 + (2.18 - 0.469i)T \)
7 \( 1 + (2.39 - 1.12i)T \)
good2 \( 1 + (0.0754 + 0.196i)T + (-1.48 + 1.33i)T^{2} \)
11 \( 1 + (0.565 + 5.38i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.378 + 2.38i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.31 + 0.852i)T + (6.91 - 15.5i)T^{2} \)
19 \( 1 + (-5.69 + 6.32i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (0.862 - 0.331i)T + (17.0 - 15.3i)T^{2} \)
29 \( 1 + (8.61 + 2.79i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.91 - 8.98i)T + (-28.3 - 12.6i)T^{2} \)
37 \( 1 + (-4.99 + 4.04i)T + (7.69 - 36.1i)T^{2} \)
41 \( 1 + (2.50 - 3.44i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (4.70 - 4.70i)T - 43iT^{2} \)
47 \( 1 + (0.296 - 0.456i)T + (-19.1 - 42.9i)T^{2} \)
53 \( 1 + (5.62 + 0.294i)T + (52.7 + 5.54i)T^{2} \)
59 \( 1 + (-3.74 + 1.66i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-0.0510 + 0.114i)T + (-40.8 - 45.3i)T^{2} \)
67 \( 1 + (5.55 + 8.56i)T + (-27.2 + 61.2i)T^{2} \)
71 \( 1 + (-0.211 + 0.649i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.85 + 2.28i)T + (-15.1 - 71.4i)T^{2} \)
79 \( 1 + (0.616 + 2.90i)T + (-72.1 + 32.1i)T^{2} \)
83 \( 1 + (4.20 - 8.24i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-5.43 - 2.41i)T + (59.5 + 66.1i)T^{2} \)
97 \( 1 + (-5.93 - 11.6i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.85358622102462453772328279973, −9.764652557035318871666333069326, −9.046681293649454906906099067333, −7.905223081561203689321044572402, −6.91190402184050986847109692310, −5.89181476124467188014823355683, −5.12581704412461038963128159327, −3.32038275885995387393118600608, −3.00129285912336357523210088984, −0.50695121219758854175402542889, 1.85011510239647348416548840313, 3.33891235620434771731481373994, 4.16340692027921541854934583879, 5.76217725869081054569123077104, 6.98126766100314601576919669404, 7.39814257201476044576585609148, 8.015139353994801989913960080465, 9.339876421843167740826989990285, 10.19304138548799901342638845164, 11.45573159505377582599520911840

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