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Properties

Label 2-525-35.33-c1-0-0
Degree $2$
Conductor $525$
Sign $0.975 + 0.222i$
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  + (−0.650 − 2.42i)2-s + (0.965 + 0.258i)3-s + (−3.73 + 2.15i)4-s − 2.51i·6-s + (−2.09 + 1.61i)7-s + (4.11 + 4.11i)8-s + (0.866 + 0.499i)9-s + (2.73 + 4.74i)11-s + (−4.17 + 1.11i)12-s + (−0.579 + 0.579i)13-s + (5.29 + 4.02i)14-s + (3.00 − 5.19i)16-s + (−1.22 + 4.58i)17-s + (0.650 − 2.42i)18-s + (−0.220 + 0.381i)19-s + ⋯
L(s)  = 1  + (−0.459 − 1.71i)2-s + (0.557 + 0.149i)3-s + (−1.86 + 1.07i)4-s − 1.02i·6-s + (−0.791 + 0.611i)7-s + (1.45 + 1.45i)8-s + (0.288 + 0.166i)9-s + (0.825 + 1.42i)11-s + (−1.20 + 0.322i)12-s + (−0.160 + 0.160i)13-s + (1.41 + 1.07i)14-s + (0.750 − 1.29i)16-s + (−0.298 + 1.11i)17-s + (0.153 − 0.572i)18-s + (−0.0505 + 0.0875i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.975 + 0.222i$
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: \((2,\ 525,\ (\ :1/2),\ 0.975 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950948 - 0.106942i\)
\(L(\frac12)\) \(\approx\) \(0.950948 - 0.106942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 1.61i)T \)
good2 \( 1 + (0.650 + 2.42i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-2.73 - 4.74i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.579 - 0.579i)T - 13iT^{2} \)
17 \( 1 + (1.22 - 4.58i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.220 - 0.381i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.70 - 0.457i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.853iT - 29T^{2} \)
31 \( 1 + (-2.32 + 1.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0668 + 0.249i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.321iT - 41T^{2} \)
43 \( 1 + (-0.631 - 0.631i)T + 43iT^{2} \)
47 \( 1 + (-7.91 + 2.12i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.96 - 11.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.89 - 5.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.73 + 3.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.16 + 1.38i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + (8.53 + 2.28i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-9.02 - 5.20i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.47 + 8.47i)T - 83iT^{2} \)
89 \( 1 + (4.03 - 6.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.99 - 5.99i)T + 97iT^{2} \)
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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

−10.60195725487380997732467208147, −9.978738602247113734962115233473, −9.264022753494942708080381951241, −8.753481221963810289373853397469, −7.55535730670828472654617383052, −6.24183954023585507439973443872, −4.50328024853624065342574586933, −3.77622782807987329217187883630, −2.59807981471866843170048149813, −1.69805705812326641162575446215, 0.63871655586499553559150780485, 3.19684611188657572633698250343, 4.40952096500713801174679461886, 5.72674126726843662211885565728, 6.55668468276816293216736202963, 7.19692082263563045820059056492, 8.123417942425337610242220912998, 8.949393724492244561446045333965, 9.491902268393504792566046255284, 10.49256242763417779588425512051

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