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Properties

Label 2-525-35.12-c1-0-11
Degree $2$
Conductor $525$
Sign $0.862 + 0.506i$
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  + (−1.67 + 0.448i)2-s + (0.258 − 0.965i)3-s + (0.866 − 0.500i)4-s + 1.73i·6-s + (2.38 − 1.15i)7-s + (1.22 − 1.22i)8-s + (−0.866 − 0.499i)9-s + (3 + 5.19i)11-s + (−0.258 − 0.965i)12-s + (−3.53 − 3.53i)13-s + (−3.46 + 3i)14-s + (−2.5 + 4.33i)16-s + (1.67 + 0.448i)18-s + (2.59 − 4.5i)19-s + (−0.500 − 2.59i)21-s + (−7.34 − 7.34i)22-s + ⋯
L(s)  = 1  + (−1.18 + 0.316i)2-s + (0.149 − 0.557i)3-s + (0.433 − 0.250i)4-s + 0.707i·6-s + (0.899 − 0.436i)7-s + (0.433 − 0.433i)8-s + (−0.288 − 0.166i)9-s + (0.904 + 1.56i)11-s + (−0.0747 − 0.278i)12-s + (−0.980 − 0.980i)13-s + (−0.925 + 0.801i)14-s + (−0.625 + 1.08i)16-s + (0.394 + 0.105i)18-s + (0.596 − 1.03i)19-s + (−0.109 − 0.566i)21-s + (−1.56 − 1.56i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840200 - 0.228434i\)
\(L(\frac12)\) \(\approx\) \(0.840200 - 0.228434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.38 + 1.15i)T \)
good2 \( 1 + (1.67 - 0.448i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.53 + 3.53i)T + 13iT^{2} \)
17 \( 1 + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.59 + 4.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.896 - 3.34i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.36 + 2.24i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (-2.44 + 2.44i)T - 43iT^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.46 - 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.24 + 8.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (1.81 - 6.76i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.52 - 5.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + (-6.92 + 12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.53 - 3.53i)T - 97iT^{2} \)
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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

−10.52489652313040745857789573829, −9.663393294089968105356648020377, −9.063045722941058785027664469574, −7.87740506491085510526282008586, −7.43729674381315655654545034991, −6.80691976671944691693832423444, −5.16963123621405445628302399331, −4.12983266061548075170513685484, −2.22264670303155947481060914458, −0.926231291420088208567483815013, 1.26081232108079384532547202181, 2.68142653829516194858922451408, 4.22826894862376314004293690529, 5.22535440454917635410539394555, 6.44504877737320428555135529349, 7.906726560575395794107556626619, 8.398309535971563860616062913211, 9.275302319375137904814802195287, 9.790828430382109411634449687257, 10.92455303349627669164667485961

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