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Properties

Label 2-525-21.20-c1-0-32
Degree $2$
Conductor $525$
Sign $0.996 + 0.0777i$
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  + 2.40i·2-s + (0.777 − 1.54i)3-s − 3.79·4-s + (3.72 + 1.87i)6-s + (−1 − 2.44i)7-s − 4.31i·8-s + (−1.79 − 2.40i)9-s − 4.31i·11-s + (−2.94 + 5.86i)12-s + 2.44i·13-s + (5.89 − 2.40i)14-s + 2.79·16-s + 5.89·17-s + (5.79 − 4.31i)18-s − 6.83i·19-s + ⋯
L(s)  = 1  + 1.70i·2-s + (0.448 − 0.893i)3-s − 1.89·4-s + (1.52 + 0.763i)6-s + (−0.377 − 0.925i)7-s − 1.52i·8-s + (−0.597 − 0.802i)9-s − 1.29i·11-s + (−0.850 + 1.69i)12-s + 0.679i·13-s + (1.57 − 0.643i)14-s + 0.697·16-s + 1.42·17-s + (1.36 − 1.01i)18-s − 1.56i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.996 + 0.0777i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.996 + 0.0777i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26149 - 0.0491371i\)
\(L(\frac12)\) \(\approx\) \(1.26149 - 0.0491371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.777 + 1.54i)T \)
5 \( 1 \)
7 \( 1 + (1 + 2.44i)T \)
good2 \( 1 - 2.40iT - 2T^{2} \)
11 \( 1 + 4.31iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 5.89T + 17T^{2} \)
19 \( 1 + 6.83iT - 19T^{2} \)
23 \( 1 - 0.502iT - 23T^{2} \)
29 \( 1 + 0.502iT - 29T^{2} \)
31 \( 1 + 4.38iT - 31T^{2} \)
37 \( 1 + 0.582T + 37T^{2} \)
41 \( 1 - 4.66T + 41T^{2} \)
43 \( 1 - 2.58T + 43T^{2} \)
47 \( 1 - 5.89T + 47T^{2} \)
53 \( 1 - 13.4iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 4.38iT - 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 4.31iT - 71T^{2} \)
73 \( 1 + 6.32iT - 73T^{2} \)
79 \( 1 + 8.58T + 79T^{2} \)
83 \( 1 - 7.12T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 0.511iT - 97T^{2} \)
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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

−10.79789266803812440958906951315, −9.341533139163775273108008021673, −8.857931149380174707194240860798, −7.68350622389258093172183016650, −7.40655232611237635905311816220, −6.35757752616565336543462871583, −5.79282178705937015412237328258, −4.35646824416733090384045890723, −3.09721076729793508277789375377, −0.73458843159729721460469424748, 1.83845549220311874046192354998, 2.95029764577893361547127771262, 3.70702912479599056044885291207, 4.84320864029410919855897078810, 5.75555554288578862053359872170, 7.69999130813348603160443946815, 8.667121877908752423247145065166, 9.570621694540924505422647783947, 10.05329208376087574231107391698, 10.60427668352238321759928510357

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