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Properties

Label 2-525-35.17-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.557 - 0.830i$
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  + (−0.965 + 0.258i)3-s + (1.73 + i)4-s + (−2.63 + 0.189i)7-s + (0.866 − 0.499i)9-s + (−3 + 5.19i)11-s + (−1.93 − 0.517i)12-s + (−1.41 − 1.41i)13-s + (1.99 + 3.46i)16-s + (1.55 + 5.79i)17-s + (−1.73 − 3i)19-s + (2.50 − 0.866i)21-s + (−0.707 + 0.707i)27-s + (−4.76 − 2.31i)28-s + (−4.5 − 2.59i)31-s + (1.55 − 5.79i)33-s + ⋯
L(s)  = 1  + (−0.557 + 0.149i)3-s + (0.866 + 0.5i)4-s + (−0.997 + 0.0716i)7-s + (0.288 − 0.166i)9-s + (−0.904 + 1.56i)11-s + (−0.557 − 0.149i)12-s + (−0.392 − 0.392i)13-s + (0.499 + 0.866i)16-s + (0.376 + 1.40i)17-s + (−0.397 − 0.688i)19-s + (0.545 − 0.188i)21-s + (−0.136 + 0.136i)27-s + (−0.899 − 0.436i)28-s + (−0.808 − 0.466i)31-s + (0.270 − 1.00i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.557 - 0.830i$
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: \((2,\ 525,\ (\ :1/2),\ -0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427784 + 0.802092i\)
\(L(\frac12)\) \(\approx\) \(0.427784 + 0.802092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.189i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (-1.55 - 5.79i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.73 + 3i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.24 - 8.36i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (-3.67 + 3.67i)T - 43iT^{2} \)
47 \( 1 + (5.79 + 1.55i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.68 - 10.0i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.19 + 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.5 + 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.34 + 0.896i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-0.965 + 0.258i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.33 - 2.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.19 + 9.19i)T - 97iT^{2} \)
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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

−11.11032522334610813561189261450, −10.24245426382331027179787895807, −9.779931909256065842739380770707, −8.324346698243193638904788689130, −7.37360758713515432111311128443, −6.67062210862637416408503932004, −5.75227279592735089331149928631, −4.52423455768890817243656155478, −3.25391789292548959715191719530, −2.07940811170143476097044867019, 0.52425848865602382620456819157, 2.41663640039743755558624071599, 3.51564662680074924093609276015, 5.33896798165960565115761826201, 5.83970684961632803735877983253, 6.85979835171943656426620616012, 7.53750383937065199442356792049, 8.882490729351416046462987399543, 9.938449096621313867284756950707, 10.61605921156715170939257648535

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