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Properties

Label 2-525-7.2-c1-0-7
Degree $2$
Conductor $525$
Sign $0.0725 - 0.997i$
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  + (−0.707 + 1.22i)2-s + (−0.5 − 0.866i)3-s + 1.41·6-s + (2.62 + 0.358i)7-s − 2.82·8-s + (−0.499 + 0.866i)9-s + (−0.292 − 0.507i)11-s + 4.41·13-s + (−2.29 + 2.95i)14-s + (2.00 − 3.46i)16-s + (1.12 + 1.94i)17-s + (−0.707 − 1.22i)18-s + (−2.32 + 4.03i)19-s + (−1 − 2.44i)21-s + 0.828·22-s + (1.12 − 1.94i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (−0.288 − 0.499i)3-s + 0.577·6-s + (0.990 + 0.135i)7-s − 0.999·8-s + (−0.166 + 0.288i)9-s + (−0.0883 − 0.152i)11-s + 1.22·13-s + (−0.612 + 0.790i)14-s + (0.500 − 0.866i)16-s + (0.271 + 0.471i)17-s + (−0.166 − 0.288i)18-s + (−0.534 + 0.925i)19-s + (−0.218 − 0.534i)21-s + 0.176·22-s + (0.233 − 0.404i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.0725 - 0.997i$
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: \((2,\ 525,\ (\ :1/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827879 + 0.769835i\)
\(L(\frac12)\) \(\approx\) \(0.827879 + 0.769835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 - 0.358i)T \)
good2 \( 1 + (0.707 - 1.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.292 + 0.507i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
17 \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.32 - 4.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + (-2.91 - 5.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.20 - 7.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 7.58T + 43T^{2} \)
47 \( 1 + (6.65 - 11.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.41 - 5.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.24 + 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.86 + 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.585T + 71T^{2} \)
73 \( 1 + (6.03 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.32 - 5.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.58T + 83T^{2} \)
89 \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−11.05069575143202926198701241012, −10.24735569669751085466697161763, −8.668627437805620255663258036597, −8.401502730669274747806900237704, −7.59680816073005120190443971702, −6.46368048800085617685875886813, −5.95355349157480781183110511691, −4.67672434575019028170946460263, −3.12751126942958120730267843995, −1.42435431579097682436193401469, 0.940784216866663612608743471700, 2.35162858068350172181695472222, 3.71232974096582585921858578820, 4.90219962096394044195593052436, 5.88870260812422690972763569435, 7.00456498950766796807564399515, 8.428564072181812511708945721613, 8.936159641532423767756882266191, 10.09383901720198949682366267427, 10.60434150829747641388565419241

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