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Properties

Label 2-525-7.3-c2-0-48
Degree $2$
Conductor $525$
Sign $-0.985 - 0.171i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.836 − 1.44i)2-s + (1.5 + 0.866i)3-s + (0.599 − 1.03i)4-s − 2.89i·6-s + (−4.76 − 5.13i)7-s − 8.70·8-s + (1.5 + 2.59i)9-s + (6.91 − 11.9i)11-s + (1.79 − 1.03i)12-s − 6.12i·13-s + (−3.45 + 11.1i)14-s + (4.88 + 8.45i)16-s + (2.14 + 1.23i)17-s + (2.51 − 4.34i)18-s + (−24.2 + 13.9i)19-s + ⋯
L(s)  = 1  + (−0.418 − 0.724i)2-s + (0.5 + 0.288i)3-s + (0.149 − 0.259i)4-s − 0.483i·6-s + (−0.680 − 0.733i)7-s − 1.08·8-s + (0.166 + 0.288i)9-s + (0.628 − 1.08i)11-s + (0.149 − 0.0865i)12-s − 0.470i·13-s + (−0.246 + 0.799i)14-s + (0.305 + 0.528i)16-s + (0.126 + 0.0729i)17-s + (0.139 − 0.241i)18-s + (−1.27 + 0.736i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.985 - 0.171i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.985 - 0.171i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9006934868\)
\(L(\frac12)\) \(\approx\) \(0.9006934868\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (4.76 + 5.13i)T \)
good2 \( 1 + (0.836 + 1.44i)T + (-2 + 3.46i)T^{2} \)
11 \( 1 + (-6.91 + 11.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 6.12iT - 169T^{2} \)
17 \( 1 + (-2.14 - 1.23i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (24.2 - 13.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.62 - 11.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 27.6T + 841T^{2} \)
31 \( 1 + (16.2 + 9.36i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (20.5 + 35.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.5iT - 1.68e3T^{2} \)
43 \( 1 + 7.60T + 1.84e3T^{2} \)
47 \( 1 + (-11.9 + 6.88i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-46.2 + 80.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (61.5 + 35.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (100. - 57.8i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (5.70 - 9.87i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 99.4T + 5.04e3T^{2} \)
73 \( 1 + (90.1 + 52.0i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (64.4 + 111. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 30.3iT - 6.88e3T^{2} \)
89 \( 1 + (-93.9 + 54.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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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

−10.40096007715754741626899691589, −9.333149362589753238992650739288, −8.798838893722522364635776964635, −7.62336890200606322396308300670, −6.44441918886197091704297693811, −5.62835071996829822949202678264, −3.88175011149880718508057262770, −3.24342761144266413238268276614, −1.81515046365651662163107298967, −0.35562841508001243629573854875, 2.05443930185927962462508569275, 3.13589152366151533150933728558, 4.45402874659706640609982343020, 6.01589634160491820393580108794, 6.77819965505660297467831886555, 7.36851841683332626592888665064, 8.578921923101040966743207643938, 9.055761803955606347355574022554, 9.803403413643076217070083585181, 11.17483678854116733986741140866

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