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Properties

Label 2-525-21.17-c1-0-22
Degree $2$
Conductor $525$
Sign $0.956 - 0.291i$
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  + (0.558 − 0.322i)2-s + (1.28 − 1.15i)3-s + (−0.792 + 1.37i)4-s + (0.346 − 1.06i)6-s + (−0.105 + 2.64i)7-s + 2.31i·8-s + (0.319 − 2.98i)9-s + (3.51 + 2.02i)11-s + (0.567 + 2.68i)12-s + 4.21i·13-s + (0.793 + 1.51i)14-s + (−0.839 − 1.45i)16-s + (1.08 − 1.88i)17-s + (−0.783 − 1.76i)18-s + (3.87 − 2.23i)19-s + ⋯
L(s)  = 1  + (0.394 − 0.227i)2-s + (0.743 − 0.668i)3-s + (−0.396 + 0.685i)4-s + (0.141 − 0.433i)6-s + (−0.0397 + 0.999i)7-s + 0.817i·8-s + (0.106 − 0.994i)9-s + (1.05 + 0.611i)11-s + (0.163 + 0.774i)12-s + 1.16i·13-s + (0.212 + 0.403i)14-s + (−0.209 − 0.363i)16-s + (0.263 − 0.457i)17-s + (−0.184 − 0.416i)18-s + (0.889 − 0.513i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.956 - 0.291i$
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: \((2,\ 525,\ (\ :1/2),\ 0.956 - 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07755 + 0.309098i\)
\(L(\frac12)\) \(\approx\) \(2.07755 + 0.309098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.28 + 1.15i)T \)
5 \( 1 \)
7 \( 1 + (0.105 - 2.64i)T \)
good2 \( 1 + (-0.558 + 0.322i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-3.51 - 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.21iT - 13T^{2} \)
17 \( 1 + (-1.08 + 1.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.87 + 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.558 + 0.322i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 + 0.195i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.13 + 3.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 + 6.54T + 43T^{2} \)
47 \( 1 + (-3.90 - 6.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.23 + 3.60i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.66 + 9.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.05 + 3.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.36 - 7.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (4.53 + 2.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.87 - 3.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.27T + 83T^{2} \)
89 \( 1 + (-0.447 - 0.774i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.89iT - 97T^{2} \)
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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

−11.46608433901703204930166974852, −9.382715786964813958409937092406, −9.283917697164850745363629098263, −8.310021490098629445545619568591, −7.30636477256612346104922904775, −6.50614231534278379862962840661, −5.09253836768562600309606901690, −3.96814718718244535098929486131, −2.94296670568015681206656329809, −1.85042108140043392624626732594, 1.19358334132533262456563698318, 3.34672576399684012220007441382, 3.98186463625474409122588885209, 5.08136373468127323142928697057, 6.01752278184106424833981264417, 7.21892798972681732955788309327, 8.224234441064085744507866555672, 9.135383216082937989857088668772, 10.10351331579463022291262653357, 10.39827196877233429289614248146

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