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Properties

Label 2-525-25.6-c1-0-6
Degree $2$
Conductor $525$
Sign $0.129 - 0.991i$
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  + (0.139 + 0.430i)2-s + (−0.809 + 0.587i)3-s + (1.45 − 1.05i)4-s + (−0.891 + 2.05i)5-s + (−0.366 − 0.266i)6-s + 7-s + (1.39 + 1.01i)8-s + (0.309 − 0.951i)9-s + (−1.00 − 0.0972i)10-s + (0.158 + 0.487i)11-s + (−0.554 + 1.70i)12-s + (−1.06 + 3.27i)13-s + (0.139 + 0.430i)14-s + (−0.483 − 2.18i)15-s + (0.868 − 2.67i)16-s + (3.91 + 2.84i)17-s + ⋯
L(s)  = 1  + (0.0989 + 0.304i)2-s + (−0.467 + 0.339i)3-s + (0.726 − 0.527i)4-s + (−0.398 + 0.917i)5-s + (−0.149 − 0.108i)6-s + 0.377·7-s + (0.491 + 0.357i)8-s + (0.103 − 0.317i)9-s + (−0.318 − 0.0307i)10-s + (0.0477 + 0.146i)11-s + (−0.160 + 0.492i)12-s + (−0.295 + 0.908i)13-s + (0.0374 + 0.115i)14-s + (−0.124 − 0.563i)15-s + (0.217 − 0.668i)16-s + (0.948 + 0.689i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.129 - 0.991i$
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: \((2,\ 525,\ (\ :1/2),\ 0.129 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11667 + 0.980784i\)
\(L(\frac12)\) \(\approx\) \(1.11667 + 0.980784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.891 - 2.05i)T \)
7 \( 1 - T \)
good2 \( 1 + (-0.139 - 0.430i)T + (-1.61 + 1.17i)T^{2} \)
11 \( 1 + (-0.158 - 0.487i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.06 - 3.27i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.91 - 2.84i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.36 + 0.992i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.50 - 4.63i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.12 - 2.26i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.02 - 2.92i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.694 + 2.13i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.00 - 9.25i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.15T + 43T^{2} \)
47 \( 1 + (1.12 - 0.817i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.52 + 1.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.84 + 11.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.04 + 3.22i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.40 + 2.47i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.93 + 2.13i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.48 - 4.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.88 + 1.36i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.16 + 1.57i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.35 + 13.4i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (0.583 - 0.424i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.20306342864600016632966088528, −10.29416169369524278118593967759, −9.608003379291320793723284820385, −8.132111842535998974660659223769, −7.21091212993146165486755444492, −6.53899622034183297542540825341, −5.61237170463509384506051334743, −4.54625925888914992973165289558, −3.24705970001386548010415619527, −1.74161346177014310785107747099, 0.957242420269928204560380259700, 2.48993433067969466378883561724, 3.85265043370401468201090784919, 4.99166631637273851686133078487, 5.94767554592489974004795314677, 7.23136421985553126585080088300, 7.86465941492033170783863163262, 8.654039083770803187587207820374, 10.01920821702352689556119120497, 10.83759047889946415759902991302

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