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Properties

Label 2-525-105.89-c1-0-34
Degree $2$
Conductor $525$
Sign $0.779 + 0.626i$
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  + (−1.12 + 1.94i)2-s + (1.42 − 0.983i)3-s + (−1.53 − 2.65i)4-s + (0.313 + 3.88i)6-s + (−2.22 − 1.42i)7-s + 2.39·8-s + (1.06 − 2.80i)9-s + (1.64 − 0.952i)11-s + (−4.79 − 2.27i)12-s − 5.07·13-s + (5.29 − 2.73i)14-s + (0.369 − 0.639i)16-s + (3.85 − 2.22i)17-s + (4.26 + 5.22i)18-s + (−3.85 − 2.22i)19-s + ⋯
L(s)  = 1  + (−0.795 + 1.37i)2-s + (0.822 − 0.568i)3-s + (−0.766 − 1.32i)4-s + (0.128 + 1.58i)6-s + (−0.841 − 0.540i)7-s + 0.846·8-s + (0.354 − 0.935i)9-s + (0.497 − 0.287i)11-s + (−1.38 − 0.656i)12-s − 1.40·13-s + (1.41 − 0.730i)14-s + (0.0923 − 0.159i)16-s + (0.936 − 0.540i)17-s + (1.00 + 1.23i)18-s + (−0.884 − 0.510i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.779 + 0.626i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.779 + 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759403 - 0.267217i\)
\(L(\frac12)\) \(\approx\) \(0.759403 - 0.267217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.42 + 0.983i)T \)
5 \( 1 \)
7 \( 1 + (2.22 + 1.42i)T \)
good2 \( 1 + (1.12 - 1.94i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.64 + 0.952i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 + (-3.85 + 2.22i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.85 + 2.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.42 - 2.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.82iT - 29T^{2} \)
31 \( 1 + (-4.81 + 2.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.02 + 2.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.250T + 41T^{2} \)
43 \( 1 + 9.23iT - 43T^{2} \)
47 \( 1 + (2.66 + 1.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.21 - 2.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.95 - 12.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.51 + 0.874i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.18 + 4.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.68iT - 71T^{2} \)
73 \( 1 + (-3.15 - 5.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.59 - 2.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.98iT - 83T^{2} \)
89 \( 1 + (5.43 - 9.41i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.94T + 97T^{2} \)
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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

−9.995849729740717813013595814257, −9.724187359158113684927020425144, −8.796185292845007545875998257286, −7.917921637835560470800720202052, −7.19171310982536686156702544057, −6.65095124580923133968889826235, −5.62030995396966763938735256666, −4.04438154694982367093580172087, −2.63306660478341607724008276358, −0.54313324691816290839295281367, 1.83811990629379902039319885981, 2.87397068545241758669890594102, 3.66114112767324681301265016036, 4.92423444721945188143188073299, 6.52563133817726925825079232867, 7.88480644617431018561315436484, 8.689122183478763115541902478332, 9.425021359590963933326678714633, 10.08876829277321553742928938434, 10.49626255354516205891743021797

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