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Properties

Label 2-525-105.89-c1-0-8
Degree $2$
Conductor $525$
Sign $0.467 - 0.884i$
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  + (0.450 − 0.780i)2-s + (−1.60 − 0.641i)3-s + (0.593 + 1.02i)4-s + (−1.22 + 0.966i)6-s + (−2.64 − 0.105i)7-s + 2.87·8-s + (2.17 + 2.06i)9-s + (−5.46 + 3.15i)11-s + (−0.295 − 2.03i)12-s + 3.77·13-s + (−1.27 + 2.01i)14-s + (0.107 − 0.185i)16-s + (−3.27 + 1.88i)17-s + (2.59 − 0.768i)18-s + (4.57 + 2.64i)19-s + ⋯
L(s)  = 1  + (0.318 − 0.551i)2-s + (−0.928 − 0.370i)3-s + (0.296 + 0.514i)4-s + (−0.500 + 0.394i)6-s + (−0.999 − 0.0398i)7-s + 1.01·8-s + (0.725 + 0.688i)9-s + (−1.64 + 0.950i)11-s + (−0.0853 − 0.587i)12-s + 1.04·13-s + (−0.340 + 0.538i)14-s + (0.0268 − 0.0464i)16-s + (−0.793 + 0.458i)17-s + (0.611 − 0.181i)18-s + (1.05 + 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.467 - 0.884i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.467 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804312 + 0.484691i\)
\(L(\frac12)\) \(\approx\) \(0.804312 + 0.484691i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.641i)T \)
5 \( 1 \)
7 \( 1 + (2.64 + 0.105i)T \)
good2 \( 1 + (-0.450 + 0.780i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (5.46 - 3.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + (3.27 - 1.88i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.57 - 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.38 - 5.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 + (-0.349 + 0.201i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.15 - 0.668i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.53T + 41T^{2} \)
43 \( 1 - 4.84iT - 43T^{2} \)
47 \( 1 + (1.68 + 0.970i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.720 + 1.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.60 + 2.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.68 - 1.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.21iT - 71T^{2} \)
73 \( 1 + (-1.25 - 2.18i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.05 + 5.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + (0.590 - 1.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.10T + 97T^{2} \)
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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

−11.04035618035788603794099566838, −10.44869785579335359366079459803, −9.609340506524785591837443073319, −7.978140247865028857949043183361, −7.41069889343666696752291125302, −6.38816370379214130096865468582, −5.42523119333522312653828762482, −4.24727041720132440207693879040, −3.08541075664076113162012030950, −1.76239404788237386802974546671, 0.54416014992250007972419467253, 2.79538109128827371094282708047, 4.27852549947121728234308838349, 5.38167654153094947081522311335, 5.99626135745719638472867940708, 6.69135716733654408382127176935, 7.74634623855530554884309639420, 9.061835459584913961698504542441, 10.08949735920607783407465859571, 10.74767951667867477803441817984

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