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Properties

Label 2-525-35.17-c1-0-3
Degree $2$
Conductor $525$
Sign $-0.986 + 0.162i$
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  + (−0.538 + 2.01i)2-s + (0.965 − 0.258i)3-s + (−2.02 − 1.16i)4-s + 2.08i·6-s + (−1.74 + 1.99i)7-s + (0.495 − 0.495i)8-s + (0.866 − 0.499i)9-s + (−0.971 + 1.68i)11-s + (−2.25 − 0.604i)12-s + (1.84 + 1.84i)13-s + (−3.06 − 4.57i)14-s + (−1.60 − 2.78i)16-s + (1.62 + 6.06i)17-s + (0.538 + 2.01i)18-s + (−1.50 − 2.61i)19-s + ⋯
L(s)  = 1  + (−0.381 + 1.42i)2-s + (0.557 − 0.149i)3-s + (−1.01 − 0.584i)4-s + 0.850i·6-s + (−0.658 + 0.752i)7-s + (0.175 − 0.175i)8-s + (0.288 − 0.166i)9-s + (−0.292 + 0.507i)11-s + (−0.651 − 0.174i)12-s + (0.511 + 0.511i)13-s + (−0.819 − 1.22i)14-s + (−0.401 − 0.695i)16-s + (0.394 + 1.47i)17-s + (0.127 + 0.474i)18-s + (−0.345 − 0.599i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.986 + 0.162i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.986 + 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0815412 - 0.998053i\)
\(L(\frac12)\) \(\approx\) \(0.0815412 - 0.998053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 - 1.99i)T \)
good2 \( 1 + (0.538 - 2.01i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (0.971 - 1.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.84 - 1.84i)T + 13iT^{2} \)
17 \( 1 + (-1.62 - 6.06i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.50 + 2.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.35 + 1.43i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.27iT - 29T^{2} \)
31 \( 1 + (5.10 + 2.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.241 + 0.901i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (6.54 - 6.54i)T - 43iT^{2} \)
47 \( 1 + (-3.75 - 1.00i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.301 + 1.12i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.06 + 5.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.50 + 3.75i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-14.3 + 3.85i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 + (6.05 - 1.62i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-14.2 + 8.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.197 + 0.197i)T + 83iT^{2} \)
89 \( 1 + (-6.08 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−11.26111332649464054611831984283, −9.956751136139030197524564496916, −9.266294764196588921990994867945, −8.414627422938998083087608759125, −7.88525336693105466886366789502, −6.65950480642430357274987874172, −6.22663377449425596259823538692, −5.08589322746228395723111280925, −3.67047498282342503199841000953, −2.16741908479106121379690003865, 0.60650925809915933973546240715, 2.23527164473182848999212278948, 3.38386014351859753515027132115, 3.93891866772240239856674184870, 5.58046649906365888523432075600, 6.93041777918288470786954557836, 8.014891888311180446397055514802, 8.918573231354605889088846786637, 9.842354174943066085337618384159, 10.27365457330051688975064932782

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