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Properties

Label 2-525-35.4-c1-0-10
Degree $2$
Conductor $525$
Sign $0.927 - 0.374i$
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  + (1.22 − 0.707i)2-s + (−0.866 − 0.5i)3-s − 1.41·6-s + (2.09 + 1.62i)7-s + 2.82i·8-s + (0.499 + 0.866i)9-s + (−1.70 + 2.95i)11-s + 1.58i·13-s + (3.70 + 0.507i)14-s + (2.00 + 3.46i)16-s + (5.40 + 3.12i)17-s + (1.22 + 0.707i)18-s + (−3.32 − 5.76i)19-s + (−0.999 − 2.44i)21-s + 4.82i·22-s + (5.40 − 3.12i)23-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.577·6-s + (0.790 + 0.612i)7-s + 0.999i·8-s + (0.166 + 0.288i)9-s + (−0.514 + 0.891i)11-s + 0.439i·13-s + (0.990 + 0.135i)14-s + (0.500 + 0.866i)16-s + (1.31 + 0.757i)17-s + (0.288 + 0.166i)18-s + (−0.763 − 1.32i)19-s + (−0.218 − 0.534i)21-s + 1.02i·22-s + (1.12 − 0.650i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84750 + 0.359310i\)
\(L(\frac12)\) \(\approx\) \(1.84750 + 0.359310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.09 - 1.62i)T \)
good2 \( 1 + (-1.22 + 0.707i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 + (-5.40 - 3.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.32 + 5.76i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.40 + 3.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.242T + 29T^{2} \)
31 \( 1 + (-0.0857 + 0.148i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.83 - 2.79i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + (-8.06 + 4.65i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.01 - 0.585i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.707 + 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.24 + 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + (-1.79 - 1.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.32 + 4.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.41iT - 83T^{2} \)
89 \( 1 + (1.87 + 3.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−11.14101564046272076603063221706, −10.44963143438651028430580240019, −9.069469756546269627492234132909, −8.196969829216712960164611608958, −7.26980689151953205177335559865, −6.01847012227127853197770430943, −4.97408591905892692046723375410, −4.50964554992756566224615845523, −2.92922241538032384074622236385, −1.82878350282593018079574466772, 0.994371052028961731873373194665, 3.33457996545625042277637564977, 4.28131955235797569726867128804, 5.46137562585700036689992013282, 5.65539223031715343106230668623, 7.06733925609863893968037279648, 7.82904178644795570360641635416, 9.037901873343752680159041350616, 10.31607953661785592235063607528, 10.60642689582011087948869217985

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