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Properties

Label 2-525-21.5-c1-0-9
Degree $2$
Conductor $525$
Sign $0.921 + 0.389i$
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  + (−1.31 − 0.757i)2-s + (−1.20 + 1.24i)3-s + (0.147 + 0.254i)4-s + (2.52 − 0.722i)6-s + (−2.64 − 0.0753i)7-s + 2.58i·8-s + (−0.102 − 2.99i)9-s + (−1.86 + 1.07i)11-s + (−0.494 − 0.123i)12-s − 3.48i·13-s + (3.41 + 2.10i)14-s + (2.25 − 3.89i)16-s + (1.78 + 3.09i)17-s + (−2.13 + 4.01i)18-s + (1.05 + 0.611i)19-s + ⋯
L(s)  = 1  + (−0.927 − 0.535i)2-s + (−0.694 + 0.719i)3-s + (0.0735 + 0.127i)4-s + (1.02 − 0.294i)6-s + (−0.999 − 0.0284i)7-s + 0.913i·8-s + (−0.0341 − 0.999i)9-s + (−0.560 + 0.323i)11-s + (−0.142 − 0.0356i)12-s − 0.965i·13-s + (0.911 + 0.561i)14-s + (0.562 − 0.974i)16-s + (0.433 + 0.751i)17-s + (−0.503 + 0.945i)18-s + (0.242 + 0.140i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.921 + 0.389i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.921 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473856 - 0.0960398i\)
\(L(\frac12)\) \(\approx\) \(0.473856 - 0.0960398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.20 - 1.24i)T \)
5 \( 1 \)
7 \( 1 + (2.64 + 0.0753i)T \)
good2 \( 1 + (1.31 + 0.757i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.48iT - 13T^{2} \)
17 \( 1 + (-1.78 - 3.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.05 - 0.611i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.31 + 0.757i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-2.75 + 1.58i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.90 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 + (-3.05 + 5.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.72 + 5.61i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 + 1.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.94 + 1.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.15 - 8.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-5.93 + 3.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.941 - 1.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.10T + 83T^{2} \)
89 \( 1 + (-0.889 + 1.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.32iT - 97T^{2} \)
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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

−10.49569918519471094737086978782, −10.08305822936673099285008962849, −9.374767124189970476458802064879, −8.449503626528867103856682263789, −7.32954083708344096922608814160, −5.94574249068115559667597361365, −5.36468984784711447362446412444, −3.94435498088437496752305732301, −2.68739729656196698192884648545, −0.69768721273776291355182743240, 0.76576061224344557506225188227, 2.73459572994812296913396064305, 4.32161657058104762484638297092, 5.80705889596629999439535968238, 6.52082231464406522414049665306, 7.36702647513150163902647548633, 8.016207124019348912563336557844, 9.179434342607154861438374497633, 9.808134218360892187432373606763, 10.78419280476975349596205607536

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