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Properties

Label 2-525-5.4-c5-0-55
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  + 3.98i·2-s − 9i·3-s + 16.0·4-s + 35.8·6-s − 49i·7-s + 191. i·8-s − 81·9-s − 608.·11-s − 144. i·12-s + 183. i·13-s + 195.·14-s − 249.·16-s + 316. i·17-s − 323. i·18-s − 396.·19-s + ⋯
L(s)  = 1  + 0.704i·2-s − 0.577i·3-s + 0.503·4-s + 0.407·6-s − 0.377i·7-s + 1.05i·8-s − 0.333·9-s − 1.51·11-s − 0.290i·12-s + 0.301i·13-s + 0.266·14-s − 0.243·16-s + 0.265i·17-s − 0.234i·18-s − 0.252·19-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

\(L(3)\) \(\approx\) \(2.019309515\)
\(L(\frac12)\) \(\approx\) \(2.019309515\)
\(L(\frac{7}{2})\) 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 \( 1 - 3.98iT - 32T^{2} \)
11 \( 1 + 608.T + 1.61e5T^{2} \)
13 \( 1 - 183. iT - 3.71e5T^{2} \)
17 \( 1 - 316. iT - 1.41e6T^{2} \)
19 \( 1 + 396.T + 2.47e6T^{2} \)
23 \( 1 + 1.88e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.61e3T + 2.05e7T^{2} \)
31 \( 1 - 9.21e3T + 2.86e7T^{2} \)
37 \( 1 - 636. iT - 6.93e7T^{2} \)
41 \( 1 - 1.57e4T + 1.15e8T^{2} \)
43 \( 1 + 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.03e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.74e4iT - 4.18e8T^{2} \)
59 \( 1 + 6.69e3T + 7.14e8T^{2} \)
61 \( 1 - 4.53e4T + 8.44e8T^{2} \)
67 \( 1 + 5.36e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.99e4T + 1.80e9T^{2} \)
73 \( 1 + 4.68e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.71e3T + 3.07e9T^{2} \)
83 \( 1 + 8.87e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.05e4T + 5.58e9T^{2} \)
97 \( 1 + 1.67e4iT - 8.58e9T^{2} \)
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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

−10.25633373584957267733633959376, −8.688538596560795517205426918727, −8.006634320723869483052724308717, −7.27882388841201286832539737796, −6.46517052968068807427376077714, −5.61878439156033483894039059388, −4.56834947623853178826137376240, −2.88734708026506738363901630452, −2.05194729677335140142966513815, −0.51516296229740005135891815512, 0.934114940871519556586373697800, 2.49121191594385414627667453874, 2.97605704873256442992432233706, 4.32486788396010320964895359645, 5.39224908551930192871397570762, 6.34852837970677902315412707082, 7.58371454070502085811308244628, 8.377286914621536235654805666616, 9.700868668594190223663232941561, 10.12721220049602580406676978671

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