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Properties

Label 2-525-35.4-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.925 - 0.379i$
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  + (−0.427 + 0.247i)2-s + (−0.866 − 0.5i)3-s + (−0.877 + 1.52i)4-s + 0.494·6-s + (1.86 + 1.87i)7-s − 1.85i·8-s + (0.499 + 0.866i)9-s + (−2.66 + 4.62i)11-s + (1.52 − 0.877i)12-s − 5.09i·13-s + (−1.26 − 0.343i)14-s + (−1.29 − 2.24i)16-s + (−0.303 − 0.175i)17-s + (−0.427 − 0.247i)18-s + (1.38 + 2.39i)19-s + ⋯
L(s)  = 1  + (−0.302 + 0.174i)2-s + (−0.499 − 0.288i)3-s + (−0.438 + 0.760i)4-s + 0.201·6-s + (0.704 + 0.709i)7-s − 0.656i·8-s + (0.166 + 0.288i)9-s + (−0.804 + 1.39i)11-s + (0.438 − 0.253i)12-s − 1.41i·13-s + (−0.337 − 0.0917i)14-s + (−0.324 − 0.561i)16-s + (−0.0735 − 0.0424i)17-s + (−0.100 − 0.0582i)18-s + (0.317 + 0.549i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.925 - 0.379i$
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: \((2,\ 525,\ (\ :1/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0937085 + 0.475813i\)
\(L(\frac12)\) \(\approx\) \(0.0937085 + 0.475813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.87i)T \)
good2 \( 1 + (0.427 - 0.247i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (2.66 - 4.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.09iT - 13T^{2} \)
17 \( 1 + (0.303 + 0.175i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.38 - 2.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.50 - 3.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (3.05 - 5.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.05 - 1.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.86T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 + (7.66 - 4.42i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.47 + 3.16i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (3.25 + 1.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (-6.64 - 3.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.18 - 2.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.87iT - 83T^{2} \)
89 \( 1 + (2.17 + 3.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.49iT - 97T^{2} \)
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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

−11.34716955647624215338641318625, −10.20831478516013547416818075171, −9.562660791344155221568773545719, −8.151884732007460517624924369741, −7.934814400855307067569185780826, −6.93760159739870542369784633824, −5.49231036875446968177084439346, −4.90444329045725408182430888313, −3.45643359408414031323207791208, −1.95505052119967522859861589267, 0.32998435000809165745965217395, 1.90744400506504890934318377216, 3.87193879019630366847326106328, 4.83124262164406627866059058222, 5.69035377379757419486319198383, 6.67348041954034412107091766964, 7.982389321692956927354449835618, 8.793333838410920831136935612404, 9.735783366705447880422345015424, 10.55890887822231742897408910303

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