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Properties

Label 2-525-5.4-c3-0-45
Degree $2$
Conductor $525$
Sign $-0.894 + 0.447i$
Analytic cond. $30.9760$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56i·2-s − 3i·3-s + 1.40·4-s − 7.70·6-s − 7i·7-s − 24.1i·8-s − 9·9-s + 66.4·11-s − 4.21i·12-s − 34.0i·13-s − 17.9·14-s − 50.7·16-s − 6.12i·17-s + 23.1i·18-s + 163.·19-s + ⋯
L(s)  = 1  − 0.908i·2-s − 0.577i·3-s + 0.175·4-s − 0.524·6-s − 0.377i·7-s − 1.06i·8-s − 0.333·9-s + 1.82·11-s − 0.101i·12-s − 0.726i·13-s − 0.343·14-s − 0.793·16-s − 0.0874i·17-s + 0.302i·18-s + 1.97·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(30.9760\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.537430279\)
\(L(\frac12)\) \(\approx\) \(2.537430279\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good2 \( 1 + 2.56iT - 8T^{2} \)
11 \( 1 - 66.4T + 1.33e3T^{2} \)
13 \( 1 + 34.0iT - 2.19e3T^{2} \)
17 \( 1 + 6.12iT - 4.91e3T^{2} \)
19 \( 1 - 163.T + 6.85e3T^{2} \)
23 \( 1 - 35.8iT - 1.21e4T^{2} \)
29 \( 1 + 27.7T + 2.43e4T^{2} \)
31 \( 1 - 74.3T + 2.97e4T^{2} \)
37 \( 1 - 260. iT - 5.06e4T^{2} \)
41 \( 1 + 445.T + 6.89e4T^{2} \)
43 \( 1 + 474. iT - 7.95e4T^{2} \)
47 \( 1 - 51.0iT - 1.03e5T^{2} \)
53 \( 1 + 676. iT - 1.48e5T^{2} \)
59 \( 1 - 115.T + 2.05e5T^{2} \)
61 \( 1 + 390.T + 2.26e5T^{2} \)
67 \( 1 - 713. iT - 3.00e5T^{2} \)
71 \( 1 - 810.T + 3.57e5T^{2} \)
73 \( 1 - 350. iT - 3.89e5T^{2} \)
79 \( 1 - 50.9T + 4.93e5T^{2} \)
83 \( 1 + 84.8iT - 5.71e5T^{2} \)
89 \( 1 + 1.52e3T + 7.04e5T^{2} \)
97 \( 1 - 1.23e3iT - 9.12e5T^{2} \)
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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

−10.06927246451730721282968612391, −9.508387246979495610425272851161, −8.325255113702668313986492114076, −7.13976555985617773319948222209, −6.65954247139801700822765063411, −5.38207283088790907611390081154, −3.82349417864404875823871011415, −3.09256068706798188299707909614, −1.60531970990812225045947878584, −0.849552193957530345526195270976, 1.53496738894292689205369990934, 3.10545311818635751354621455551, 4.33161501397311897851966839648, 5.41051153143753523142176957161, 6.31915576659505043150854395085, 7.03534121972052116256251940340, 8.088487377706137440054727998947, 9.129869172533772578648662332069, 9.561156342291490723841049773071, 10.95227016476787931646772564345

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