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Properties

Label 2-525-35.33-c1-0-6
Degree $2$
Conductor $525$
Sign $-0.625 + 0.780i$
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  + (−0.698 − 2.60i)2-s + (−0.965 − 0.258i)3-s + (−4.57 + 2.64i)4-s + 2.69i·6-s + (2.58 − 0.543i)7-s + (6.26 + 6.26i)8-s + (0.866 + 0.499i)9-s + (1.69 + 2.93i)11-s + (5.10 − 1.36i)12-s + (4.01 − 4.01i)13-s + (−3.22 − 6.36i)14-s + (6.67 − 11.5i)16-s + (−1.00 + 3.76i)17-s + (0.698 − 2.60i)18-s + (0.721 − 1.24i)19-s + ⋯
L(s)  = 1  + (−0.493 − 1.84i)2-s + (−0.557 − 0.149i)3-s + (−2.28 + 1.32i)4-s + 1.10i·6-s + (0.978 − 0.205i)7-s + (2.21 + 2.21i)8-s + (0.288 + 0.166i)9-s + (0.511 + 0.885i)11-s + (1.47 − 0.394i)12-s + (1.11 − 1.11i)13-s + (−0.862 − 1.70i)14-s + (1.66 − 2.89i)16-s + (−0.244 + 0.911i)17-s + (0.164 − 0.614i)18-s + (0.165 − 0.286i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410853 - 0.855276i\)
\(L(\frac12)\) \(\approx\) \(0.410853 - 0.855276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.58 + 0.543i)T \)
good2 \( 1 + (0.698 + 2.60i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-1.69 - 2.93i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.01 + 4.01i)T - 13iT^{2} \)
17 \( 1 + (1.00 - 3.76i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.721 + 1.24i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.739 + 0.198i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.95iT - 29T^{2} \)
31 \( 1 + (-3.17 + 1.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.325 + 1.21i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 1.05iT - 41T^{2} \)
43 \( 1 + (6.74 + 6.74i)T + 43iT^{2} \)
47 \( 1 + (-6.55 + 1.75i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.94 + 10.9i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.22 - 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.9 - 6.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.544 + 0.145i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.73T + 71T^{2} \)
73 \( 1 + (4.78 + 1.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.79 + 3.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.49 - 1.49i)T - 83iT^{2} \)
89 \( 1 + (-1.78 + 3.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.36 - 1.36i)T + 97iT^{2} \)
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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

−10.48312803974658490020817623761, −10.24702434131514917647385443873, −8.853075359765698839745555874132, −8.324163115546134430160113291640, −7.18533238102221888306422763698, −5.49494772495923445180652480946, −4.46265746691330295199696000357, −3.56295854243181905374901117280, −2.01195168298697371950813831045, −1.02182789756860882407340548884, 1.11769888805362937707904469519, 4.08770659720949049035875347647, 4.92800448613352517942648618601, 5.92562874836304990342606481710, 6.50913654280989169184107989048, 7.51012432418932046533254782316, 8.481637107546042621666543833264, 8.989484477093873382967602566250, 9.951264601355183471472375610711, 11.15791523662582091773871702511

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