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Properties

Label 2-525-105.53-c1-0-8
Degree $2$
Conductor $525$
Sign $-0.648 - 0.761i$
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  + (−0.582 + 2.17i)2-s + (−1.71 + 0.245i)3-s + (−2.64 − 1.52i)4-s + (0.465 − 3.86i)6-s + (1.15 − 2.38i)7-s + (1.68 − 1.68i)8-s + (2.87 − 0.840i)9-s + (3.88 + 2.24i)11-s + (4.91 + 1.97i)12-s + (1.08 + 1.08i)13-s + (4.50 + 3.88i)14-s + (−0.381 − 0.660i)16-s + (2.04 − 0.548i)17-s + (0.150 + 6.74i)18-s + (−3.66 + 2.11i)19-s + ⋯
L(s)  = 1  + (−0.411 + 1.53i)2-s + (−0.989 + 0.141i)3-s + (−1.32 − 0.764i)4-s + (0.189 − 1.57i)6-s + (0.435 − 0.900i)7-s + (0.595 − 0.595i)8-s + (0.959 − 0.280i)9-s + (1.17 + 0.676i)11-s + (1.41 + 0.569i)12-s + (0.300 + 0.300i)13-s + (1.20 + 1.03i)14-s + (−0.0952 − 0.165i)16-s + (0.496 − 0.133i)17-s + (0.0354 + 1.58i)18-s + (−0.839 + 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361098 + 0.782170i\)
\(L(\frac12)\) \(\approx\) \(0.361098 + 0.782170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.245i)T \)
5 \( 1 \)
7 \( 1 + (-1.15 + 2.38i)T \)
good2 \( 1 + (0.582 - 2.17i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-3.88 - 2.24i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.08 - 1.08i)T + 13iT^{2} \)
17 \( 1 + (-2.04 + 0.548i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.66 - 2.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.13 - 0.840i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + (-0.530 + 0.918i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.75 - 1.54i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.84iT - 41T^{2} \)
43 \( 1 + (2.00 + 2.00i)T + 43iT^{2} \)
47 \( 1 + (1.36 - 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.23 - 8.34i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.35 + 4.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.88 - 6.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.152 - 0.569i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.66iT - 71T^{2} \)
73 \( 1 + (-4.22 + 1.13i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.78 + 3.33i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.0 - 11.0i)T - 83iT^{2} \)
89 \( 1 + (-1.75 - 3.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.60 + 5.60i)T - 97iT^{2} \)
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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

−11.12229783820986921121228294214, −10.05776976283286997965563954073, −9.371066918027060007564464464268, −8.251546142559631494381838697933, −7.29846097890232963476349591310, −6.68741824039036327821637625752, −5.95154864327483528980860186984, −4.78682950719566872660021251764, −4.09017285716180243190290257374, −1.17470355447169958480115751218, 0.856829014403081362092767695709, 2.07098496844522106136437249673, 3.48742778613091361243970165454, 4.63220283755806403076042215573, 5.81365365520453896238301981401, 6.71745179920193953066605738293, 8.297597504799546642864798023392, 8.995260911698862891264506619996, 9.907781062322660397522497948643, 10.86009897645038745845979376943

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