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Properties

Label 2-525-105.2-c1-0-4
Degree $2$
Conductor $525$
Sign $-0.810 + 0.585i$
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 + (−1.60 − 0.644i)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 + (2.16 + 2.07i)9-s + (−3.88 + 2.24i)11-s + (5.24 − 0.750i)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 + (−3.24 + 5.91i)18-s + (−3.66 − 2.11i)19-s + ⋯
L(s)  = 1  + (0.411 + 1.53i)2-s + (−0.928 − 0.372i)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 + (0.722 + 0.691i)9-s + (−1.17 + 0.676i)11-s + (1.51 − 0.216i)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 + (−0.764 + 1.39i)18-s + (−0.839 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.810 + 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253858 - 0.785015i\)
\(L(\frac12)\) \(\approx\) \(0.253858 - 0.785015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.644i)T \)
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

−11.40022898015602830949374512298, −10.67051157191975512007883551100, −9.395646151692585662038546732620, −8.164101055313314530074127533061, −7.72125982082704401524582397941, −6.62697454435178213166597139972, −5.93402223958624804865735509037, −5.10911975145980213497451666262, −4.47324133679462186700082954923, −2.23630834542564764980186914726, 0.46245562983271310462107915349, 1.99569121734843936327197860064, 3.58531612567312926777182713480, 4.33436843459754032772508682370, 5.22593613371509757599959808264, 6.36060550740883888249619428304, 7.66431207658734390462114458414, 8.891960152286755163457663213916, 10.27198820484487324216913969505, 10.36465901570062551417614583764

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