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Properties

Label 2-525-7.2-c1-0-20
Degree $2$
Conductor $525$
Sign $-0.658 + 0.752i$
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  + (−0.247 + 0.427i)2-s + (−0.5 − 0.866i)3-s + (0.877 + 1.52i)4-s + 0.494·6-s + (−1.87 − 1.86i)7-s − 1.85·8-s + (−0.499 + 0.866i)9-s + (−2.66 − 4.62i)11-s + (0.877 − 1.52i)12-s − 5.09·13-s + (1.26 − 0.343i)14-s + (−1.29 + 2.24i)16-s + (0.175 + 0.303i)17-s + (−0.247 − 0.427i)18-s + (−1.38 + 2.39i)19-s + ⋯
L(s)  = 1  + (−0.174 + 0.302i)2-s + (−0.288 − 0.499i)3-s + (0.438 + 0.760i)4-s + 0.201·6-s + (−0.709 − 0.704i)7-s − 0.656·8-s + (−0.166 + 0.288i)9-s + (−0.804 − 1.39i)11-s + (0.253 − 0.438i)12-s − 1.41·13-s + (0.337 − 0.0917i)14-s + (−0.324 + 0.561i)16-s + (0.0424 + 0.0735i)17-s + (−0.0582 − 0.100i)18-s + (−0.317 + 0.549i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158379 - 0.348757i\)
\(L(\frac12)\) \(\approx\) \(0.158379 - 0.348757i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.87 + 1.86i)T \)
good2 \( 1 + (0.247 - 0.427i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.66 + 4.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.09T + 13T^{2} \)
17 \( 1 + (-0.175 - 0.303i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.38 - 2.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.75 + 6.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (3.05 + 5.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.76 - 3.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.86T + 41T^{2} \)
43 \( 1 + 1.41T + 43T^{2} \)
47 \( 1 + (4.42 - 7.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.16 + 5.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.42 + 11.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 3.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (-3.83 - 6.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.18 - 2.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.87T + 83T^{2} \)
89 \( 1 + (-2.17 + 3.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.49T + 97T^{2} \)
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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

−10.66272660681563337068480873858, −9.679393315976800722254221528300, −8.378594812985025671366987282802, −7.85967981778367292127371965097, −6.83553803343394770001692225612, −6.29124149108245432308822125054, −4.97948452870017243344668062067, −3.45982667972087865516506072162, −2.53820311403884394836538747814, −0.22064009339055906104822089266, 2.07748712294818835926854720810, 3.07421443257312730246516897266, 4.93015245352928165537338161380, 5.33474444562388090288782016493, 6.66705662935668265241711668563, 7.33876757869661089243812276342, 8.941280933592750040684922277993, 9.712529094329890708846125654065, 10.12456950507142903035304452012, 11.01508445830216048038833095739

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