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Properties

Label 2-525-5.4-c3-0-33
Degree $2$
Conductor $525$
Sign $0.447 - 0.894i$
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  + 5i·2-s + 3i·3-s − 17·4-s − 15·6-s + 7i·7-s − 45i·8-s − 9·9-s + 12·11-s − 51i·12-s − 30i·13-s − 35·14-s + 89·16-s − 134i·17-s − 45i·18-s + 92·19-s + ⋯
L(s)  = 1  + 1.76i·2-s + 0.577i·3-s − 2.12·4-s − 1.02·6-s + 0.377i·7-s − 1.98i·8-s − 0.333·9-s + 0.328·11-s − 1.22i·12-s − 0.640i·13-s − 0.668·14-s + 1.39·16-s − 1.91i·17-s − 0.589i·18-s + 1.11·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
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: \((2,\ 525,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.166728381\)
\(L(\frac12)\) \(\approx\) \(1.166728381\)
\(L(\frac{5}{2})\) 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 \( 1 - 5iT - 8T^{2} \)
11 \( 1 - 12T + 1.33e3T^{2} \)
13 \( 1 + 30iT - 2.19e3T^{2} \)
17 \( 1 + 134iT - 4.91e3T^{2} \)
19 \( 1 - 92T + 6.85e3T^{2} \)
23 \( 1 + 112iT - 1.21e4T^{2} \)
29 \( 1 - 58T + 2.43e4T^{2} \)
31 \( 1 + 224T + 2.97e4T^{2} \)
37 \( 1 + 146iT - 5.06e4T^{2} \)
41 \( 1 - 18T + 6.89e4T^{2} \)
43 \( 1 + 340iT - 7.95e4T^{2} \)
47 \( 1 - 208iT - 1.03e5T^{2} \)
53 \( 1 - 754iT - 1.48e5T^{2} \)
59 \( 1 + 380T + 2.05e5T^{2} \)
61 \( 1 - 718T + 2.26e5T^{2} \)
67 \( 1 - 412iT - 3.00e5T^{2} \)
71 \( 1 + 960T + 3.57e5T^{2} \)
73 \( 1 + 1.06e3iT - 3.89e5T^{2} \)
79 \( 1 + 896T + 4.93e5T^{2} \)
83 \( 1 + 436iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + 702iT - 9.12e5T^{2} \)
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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

−10.25933895455305968310546091819, −9.227145245004230714359924725357, −8.887350438553412334698872153535, −7.66615696543583308656028630782, −7.08702854628003906995801771606, −5.88921960399178111643206010912, −5.24910831615715771093284791001, −4.38375148537717703945141083609, −2.97792479975793590736842334955, −0.40926267412199916161492296753, 1.18480633670555135579211835896, 1.89776166478418392284370755303, 3.32461736289031074119610053593, 4.04907078036679444790608634026, 5.34578897225955341319423305355, 6.63258564425627448251898906408, 7.85908873739547548724140952246, 8.809394589780622704061818389405, 9.674178732454685000049996961408, 10.42595755866739323090356494508

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