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Properties

Label 2-525-105.23-c1-0-13
Degree $2$
Conductor $525$
Sign $-0.267 - 0.963i$
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  + (0.907 − 0.243i)2-s + (0.315 + 1.70i)3-s + (−0.967 + 0.558i)4-s + (0.700 + 1.46i)6-s + (2.64 + 0.0144i)7-s + (−2.07 + 2.07i)8-s + (−2.80 + 1.07i)9-s + (−0.630 + 0.363i)11-s + (−1.25 − 1.47i)12-s + (1.44 + 1.44i)13-s + (2.40 − 0.630i)14-s + (−0.257 + 0.446i)16-s + (−1.90 + 7.09i)17-s + (−2.28 + 1.65i)18-s + (0.664 + 0.383i)19-s + ⋯
L(s)  = 1  + (0.641 − 0.171i)2-s + (0.182 + 0.983i)3-s + (−0.483 + 0.279i)4-s + (0.285 + 0.599i)6-s + (0.999 + 0.00544i)7-s + (−0.732 + 0.732i)8-s + (−0.933 + 0.357i)9-s + (−0.189 + 0.109i)11-s + (−0.362 − 0.425i)12-s + (0.400 + 0.400i)13-s + (0.642 − 0.168i)14-s + (−0.0644 + 0.111i)16-s + (−0.460 + 1.71i)17-s + (−0.537 + 0.390i)18-s + (0.152 + 0.0879i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.267 - 0.963i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04362 + 1.37304i\)
\(L(\frac12)\) \(\approx\) \(1.04362 + 1.37304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.315 - 1.70i)T \)
5 \( 1 \)
7 \( 1 + (-2.64 - 0.0144i)T \)
good2 \( 1 + (-0.907 + 0.243i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.630 - 0.363i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 + (1.90 - 7.09i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.664 - 0.383i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.840 + 3.13i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.07T + 29T^{2} \)
31 \( 1 + (0.209 + 0.363i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.63 - 6.08i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.44iT - 41T^{2} \)
43 \( 1 + (-5.15 - 5.15i)T + 43iT^{2} \)
47 \( 1 + (-6.79 + 1.82i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.26 - 1.41i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.807 + 1.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.78 + 8.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.90 - 1.84i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.06iT - 71T^{2} \)
73 \( 1 + (-4.08 + 15.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.80 - 3.35i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 + 1.83i)T - 83iT^{2} \)
89 \( 1 + (-6.94 + 12.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.62 - 5.62i)T - 97iT^{2} \)
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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

−11.09353281001834791883339638766, −10.42569114053096839978859536596, −9.233447899922295569447199888894, −8.519512930706565089760552035424, −7.898541802315241514557179590102, −6.13554089634763450925737983968, −5.20523965635055414576846755411, −4.31675118697174668724952805210, −3.71488864048435445274042937209, −2.24369316929744813077572983757, 0.854813692077121321777132281108, 2.49304177272736105053057233712, 3.88174100508293958917294628219, 5.16853787302132836252919836225, 5.73491101481121511205179785209, 6.98712792797673074177640960457, 7.74954684623960768392997720340, 8.780195692221452692751825185499, 9.467675009789311871348417156611, 10.87598882173229551044465062360

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