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Properties

Label 2-525-105.104-c1-0-25
Degree $2$
Conductor $525$
Sign $0.462 - 0.886i$
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  + 2.52·2-s + (−0.396 + 1.68i)3-s + 4.37·4-s + (−1 + 4.25i)6-s + (−1.73 + 2i)7-s + 5.98·8-s + (−2.68 − 1.33i)9-s + 0.792i·11-s + (−1.73 + 7.37i)12-s + 5.84·13-s + (−4.37 + 5.04i)14-s + 6.37·16-s + 1.37i·17-s + (−6.78 − 3.37i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  + 1.78·2-s + (−0.228 + 0.973i)3-s + 2.18·4-s + (−0.408 + 1.73i)6-s + (−0.654 + 0.755i)7-s + 2.11·8-s + (−0.895 − 0.445i)9-s + 0.238i·11-s + (−0.500 + 2.12i)12-s + 1.61·13-s + (−1.16 + 1.34i)14-s + 1.59·16-s + 0.332i·17-s + (−1.59 − 0.794i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.462 - 0.886i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01145 + 1.82560i\)
\(L(\frac12)\) \(\approx\) \(3.01145 + 1.82560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.396 - 1.68i)T \)
5 \( 1 \)
7 \( 1 + (1.73 - 2i)T \)
good2 \( 1 - 2.52T + 2T^{2} \)
11 \( 1 - 0.792iT - 11T^{2} \)
13 \( 1 - 5.84T + 13T^{2} \)
17 \( 1 - 1.37iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 1.87T + 23T^{2} \)
29 \( 1 + 4.25iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 4.74iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6.74iT - 43T^{2} \)
47 \( 1 - 7.37iT - 47T^{2} \)
53 \( 1 - 8.51T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 6.74iT - 67T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 3.37T + 79T^{2} \)
83 \( 1 - 5.48iT - 83T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 - 1.08T + 97T^{2} \)
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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

−11.29042398691636254406618617963, −10.48605212502757956766534842306, −9.378659427719003252893587238769, −8.402791675978339049328229947700, −6.76210601061864801120417951788, −5.97504713522087336757521489818, −5.41831878109875559916158976110, −4.21288468558257730661323569360, −3.56734604533631380935670044955, −2.46843248416150086766864586030, 1.48074031906058615434276580132, 3.08155152125132839770837710761, 3.81768002288004657710250210751, 5.15490607658184876517176009063, 6.15996222374886863260264574738, 6.58756459680678820026569763402, 7.54725124832821133783842950243, 8.643613659140808285717868398853, 10.34821116104309978350821759375, 11.08445108835143518675643762064

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