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Properties

Label 2-525-105.89-c1-0-40
Degree $2$
Conductor $525$
Sign $-0.669 - 0.743i$
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 + (−1.56 + 0.742i)3-s + (−1.53 − 2.65i)4-s + (−0.313 + 3.88i)6-s + (−2.22 − 1.42i)7-s − 2.39·8-s + (1.89 − 2.32i)9-s + (−1.64 + 0.952i)11-s + (4.36 + 3.01i)12-s − 5.07·13-s + (−5.29 + 2.73i)14-s + (0.369 − 0.639i)16-s + (−3.85 + 2.22i)17-s + (−2.39 − 6.31i)18-s + (−3.85 − 2.22i)19-s + ⋯
L(s)  = 1  + (0.795 − 1.37i)2-s + (−0.903 + 0.428i)3-s + (−0.766 − 1.32i)4-s + (−0.128 + 1.58i)6-s + (−0.841 − 0.540i)7-s − 0.846·8-s + (0.632 − 0.774i)9-s + (−0.497 + 0.287i)11-s + (1.26 + 0.870i)12-s − 1.40·13-s + (−1.41 + 0.730i)14-s + (0.0923 − 0.159i)16-s + (−0.936 + 0.540i)17-s + (−0.564 − 1.48i)18-s + (−0.884 − 0.510i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.669 - 0.743i$
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: \((2,\ 525,\ (\ :1/2),\ -0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.224154 + 0.503418i\)
\(L(\frac12)\) \(\approx\) \(0.224154 + 0.503418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.56 - 0.742i)T \)
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

−10.56814635632662994779546559521, −9.960758189152610248492739637971, −9.060331255873962588478087826348, −7.25856243743030407201925196839, −6.40813896288247940017889742123, −5.06087243195877708265425950955, −4.51628349816999325830849362629, −3.46175612401356227774896075430, −2.21291445417934382594340789420, −0.25966244456655360609394336908, 2.56569896763764198158181076078, 4.33035133821161885593218318575, 5.14242273762914212235776217874, 6.00665686918301704282040146272, 6.64423292898930019643890786725, 7.43470857661026370285661548579, 8.317612908336727907848935199432, 9.611933514961612912663992410211, 10.55486927078273682836561624369, 11.78600658513773335094372056459

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