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Properties

Label 2-525-5.4-c5-0-54
Degree $2$
Conductor $525$
Sign $0.894 - 0.447i$
Analytic cond. $84.2015$
Root an. cond. $9.17613$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.14i·2-s + 9i·3-s − 19.0·4-s − 64.3·6-s + 49i·7-s + 92.5i·8-s − 81·9-s − 529.·11-s − 171. i·12-s − 1.14e3i·13-s − 350.·14-s − 1.27e3·16-s + 1.11e3i·17-s − 578. i·18-s − 169.·19-s + ⋯
L(s)  = 1  + 1.26i·2-s + 0.577i·3-s − 0.595·4-s − 0.729·6-s + 0.377i·7-s + 0.511i·8-s − 0.333·9-s − 1.32·11-s − 0.343i·12-s − 1.88i·13-s − 0.477·14-s − 1.24·16-s + 0.939i·17-s − 0.420i·18-s − 0.107·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(84.2015\)
Root analytic conductor: \(9.17613\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :5/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9142386437\)
\(L(\frac12)\) \(\approx\) \(0.9142386437\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good2 \( 1 - 7.14iT - 32T^{2} \)
11 \( 1 + 529.T + 1.61e5T^{2} \)
13 \( 1 + 1.14e3iT - 3.71e5T^{2} \)
17 \( 1 - 1.11e3iT - 1.41e6T^{2} \)
19 \( 1 + 169.T + 2.47e6T^{2} \)
23 \( 1 - 948. iT - 6.43e6T^{2} \)
29 \( 1 - 3.26e3T + 2.05e7T^{2} \)
31 \( 1 + 2.20e3T + 2.86e7T^{2} \)
37 \( 1 + 66.9iT - 6.93e7T^{2} \)
41 \( 1 + 9.01e3T + 1.15e8T^{2} \)
43 \( 1 + 4.43e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.61e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.45e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.90e4T + 7.14e8T^{2} \)
61 \( 1 - 3.72e4T + 8.44e8T^{2} \)
67 \( 1 + 4.65e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.95e4T + 1.80e9T^{2} \)
73 \( 1 + 7.16e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.01e4T + 3.07e9T^{2} \)
83 \( 1 + 5.08e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.20e5T + 5.58e9T^{2} \)
97 \( 1 + 1.25e5iT - 8.58e9T^{2} \)
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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

−10.31770004360358165676120661574, −8.894517562014545752712408438632, −8.141441803412383990846163131548, −7.62293134396077913006331787896, −6.33184295245406720373876216198, −5.44974241085706121292797203376, −5.04458557610479813960066878500, −3.44661155058864436051179990744, −2.33148844083656623959586195387, −0.23423928429574645888724322563, 0.924973530722190663357820411004, 2.05607382475955297268413901274, 2.79086312439363875614255000196, 4.05377885045608490278586170244, 5.04803107720463979006981886871, 6.58166926622583567486932086472, 7.18499081323116890955256666035, 8.358515646971989228429799024139, 9.404114757322246982550954288122, 10.14087515047214720414140439701

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