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Properties

Label 2-525-7.2-c1-0-4
Degree $2$
Conductor $525$
Sign $-0.968 + 0.250i$
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 + 1.73i)2-s + (0.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s − 1.99·6-s + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (3 + 5.19i)11-s + (1 − 1.73i)12-s + 3·13-s + (−5 − 1.73i)14-s + (1.99 − 3.46i)16-s + (−2 − 3.46i)17-s + (−0.999 − 1.73i)18-s + (−0.5 + 0.866i)19-s + (−2 + 1.73i)21-s − 12·22-s + ⋯
L(s)  = 1  + (−0.707 + 1.22i)2-s + (0.288 + 0.499i)3-s + (−0.499 − 0.866i)4-s − 0.816·6-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.904 + 1.56i)11-s + (0.288 − 0.499i)12-s + 0.832·13-s + (−1.33 − 0.462i)14-s + (0.499 − 0.866i)16-s + (−0.485 − 0.840i)17-s + (−0.235 − 0.408i)18-s + (−0.114 + 0.198i)19-s + (−0.436 + 0.377i)21-s − 2.55·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126143 - 0.989880i\)
\(L(\frac12)\) \(\approx\) \(0.126143 - 0.989880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (1 - 1.73i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.33015871726882319537788612982, −9.896477586106971616222271580205, −9.296518728297135060323926999916, −8.779706348119115643861110346472, −7.75420697950279746191706828978, −6.95826339055245779963999401317, −5.96171572878653631540120498057, −5.05407411604641121321106838386, −3.77254960248745380862309312713, −2.08269382618476784806709595124, 0.74154130201351039462854893281, 1.79706694307317425608173829921, 3.33679664056407299823865250386, 3.97996446654098130091831115625, 5.96571637295127254453832565960, 6.75133300900426866869539919183, 8.244995231051189764923982848980, 8.566069679677693509876507744596, 9.570936664165491272091133240933, 10.60326828591533869143859684439

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