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Properties

Label 2-525-105.2-c1-0-27
Degree $2$
Conductor $525$
Sign $-0.196 + 0.980i$
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)3-s + (1.73 − i)4-s + (−2.63 + 0.189i)7-s + (2.59 + 1.50i)9-s + (−3.34 + 0.896i)12-s + (4.89 − 4.89i)13-s + (1.99 − 3.46i)16-s + (−6.92 − 4i)19-s + (4.50 + 0.866i)21-s + (−3.67 − 3.67i)27-s + (−4.38 + 2.96i)28-s + (−3.5 − 6.06i)31-s + 6·36-s + (−5.01 + 1.34i)37-s + (−10.3 + 6i)39-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.866 − 0.5i)4-s + (−0.997 + 0.0716i)7-s + (0.866 + 0.5i)9-s + (−0.965 + 0.258i)12-s + (1.35 − 1.35i)13-s + (0.499 − 0.866i)16-s + (−1.58 − 0.917i)19-s + (0.981 + 0.188i)21-s + (−0.707 − 0.707i)27-s + (−0.827 + 0.560i)28-s + (−0.628 − 1.08i)31-s + 36-s + (−0.825 + 0.221i)37-s + (−1.66 + 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.196 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.645760 - 0.787748i\)
\(L(\frac12)\) \(\approx\) \(0.645760 - 0.787748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.67 + 0.448i)T \)
5 \( 1 \)
7 \( 1 + (2.63 - 0.189i)T \)
good2 \( 1 + (-1.73 + i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.89 + 4.89i)T - 13iT^{2} \)
17 \( 1 + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.92 + 4i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.01 - 1.34i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-8.57 + 8.57i)T - 43iT^{2} \)
47 \( 1 + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.896 - 3.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.67 - 0.448i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-14.7 - 8.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.67 - 3.67i)T + 97iT^{2} \)
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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

−10.71686051944442543593656857961, −10.13584871802992083760954806891, −8.901289418859451146109941412181, −7.64490043813028522172250419660, −6.65007383296430565309991426093, −6.10389296028357648646437164232, −5.35272006867926374366156541412, −3.77188898733229477127328568378, −2.32917428890465986019531632586, −0.66391307169393212550876107013, 1.74101269408170442835444652343, 3.48839827825580659420705717326, 4.25310315379480467698331858439, 5.95740945512718258299594728921, 6.42081690376487662753885334574, 7.13790431319859670237957735610, 8.495336749470451492265543916027, 9.426020124534598480818860776845, 10.64666405995400625471262807903, 10.91463230544079212922860525878

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