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Properties

Label 2-525-21.17-c1-0-11
Degree $2$
Conductor $525$
Sign $-0.489 - 0.871i$
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  + (−1.46 + 0.846i)2-s + (1.73 − 0.0718i)3-s + (0.433 − 0.750i)4-s + (−2.47 + 1.57i)6-s + (−1.71 + 2.01i)7-s − 1.91i·8-s + (2.98 − 0.248i)9-s + (0.399 + 0.230i)11-s + (0.695 − 1.32i)12-s + 3.38i·13-s + (0.803 − 4.40i)14-s + (2.49 + 4.31i)16-s + (−2.75 + 4.76i)17-s + (−4.17 + 2.89i)18-s + (3.49 − 2.01i)19-s + ⋯
L(s)  = 1  + (−1.03 + 0.598i)2-s + (0.999 − 0.0415i)3-s + (0.216 − 0.375i)4-s + (−1.01 + 0.641i)6-s + (−0.647 + 0.762i)7-s − 0.678i·8-s + (0.996 − 0.0829i)9-s + (0.120 + 0.0695i)11-s + (0.200 − 0.383i)12-s + 0.938i·13-s + (0.214 − 1.17i)14-s + (0.622 + 1.07i)16-s + (−0.667 + 1.15i)17-s + (−0.983 + 0.682i)18-s + (0.801 − 0.462i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.489 - 0.871i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.489 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486050 + 0.830270i\)
\(L(\frac12)\) \(\approx\) \(0.486050 + 0.830270i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0718i)T \)
5 \( 1 \)
7 \( 1 + (1.71 - 2.01i)T \)
good2 \( 1 + (1.46 - 0.846i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-0.399 - 0.230i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.38iT - 13T^{2} \)
17 \( 1 + (2.75 - 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.49 + 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.90 - 2.25i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.71iT - 29T^{2} \)
31 \( 1 + (3.01 + 1.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.89 - 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.25T + 41T^{2} \)
43 \( 1 - 8.35T + 43T^{2} \)
47 \( 1 + (1.57 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.0 + 5.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 - 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.90 - 3.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.458 - 0.793i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.52iT - 71T^{2} \)
73 \( 1 + (6.75 + 3.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.58 - 6.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.5T + 83T^{2} \)
89 \( 1 + (-1.35 - 2.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.44iT - 97T^{2} \)
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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

−10.80770393539636107066085765294, −9.697513664857349004716288022410, −9.222554393586909434032045705487, −8.645633084380178875078389078367, −7.72347537432153206168331838340, −6.89354750314935330750459955768, −6.05709585216418967124773716644, −4.29893364726177936603498242989, −3.22425183614093286621950477617, −1.74950580766077526432704933288, 0.72609227626148214806139097716, 2.32568711360185095610722266091, 3.32140658037935384353172303326, 4.57956840621935429788214679204, 6.10596958802545333397287562851, 7.59518636863631924375750043078, 7.83866235575347965902886757612, 9.203644924479167629241097168457, 9.498504321311657239917580131477, 10.35779647238373607173546480122

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