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Properties

Label 2-525-21.5-c1-0-5
Degree $2$
Conductor $525$
Sign $0.299 - 0.953i$
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  + (−2.17 − 1.25i)2-s + (−1.73 − 0.0401i)3-s + (2.14 + 3.71i)4-s + (3.71 + 2.25i)6-s + (2.06 + 1.65i)7-s − 5.74i·8-s + (2.99 + 0.138i)9-s + (−1.48 + 0.859i)11-s + (−3.56 − 6.51i)12-s + 0.360i·13-s + (−2.39 − 6.18i)14-s + (−2.91 + 5.04i)16-s + (1.27 + 2.20i)17-s + (−6.33 − 4.05i)18-s + (−4.93 − 2.84i)19-s + ⋯
L(s)  = 1  + (−1.53 − 0.886i)2-s + (−0.999 − 0.0231i)3-s + (1.07 + 1.85i)4-s + (1.51 + 0.922i)6-s + (0.778 + 0.627i)7-s − 2.03i·8-s + (0.998 + 0.0463i)9-s + (−0.448 + 0.259i)11-s + (−1.02 − 1.88i)12-s + 0.100i·13-s + (−0.639 − 1.65i)14-s + (−0.727 + 1.26i)16-s + (0.308 + 0.534i)17-s + (−1.49 − 0.956i)18-s + (−1.13 − 0.653i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.299 - 0.953i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.299 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238925 + 0.175355i\)
\(L(\frac12)\) \(\approx\) \(0.238925 + 0.175355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0401i)T \)
5 \( 1 \)
7 \( 1 + (-2.06 - 1.65i)T \)
good2 \( 1 + (2.17 + 1.25i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (1.48 - 0.859i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.360iT - 13T^{2} \)
17 \( 1 + (-1.27 - 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.93 + 2.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.17 + 1.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.76iT - 29T^{2} \)
31 \( 1 + (2.41 - 1.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.65 - 2.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.63T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.25 - 0.727i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.42 + 5.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.38 - 0.801i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.24 + 2.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + (10.0 - 5.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.93 - 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + (6.10 - 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.18iT - 97T^{2} \)
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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

−10.92936681522411340167142449667, −10.36071893428212586258113839829, −9.401907903596192569635708465471, −8.521522795922991547519577350073, −7.73943979401607987703174903840, −6.74332070929337486906181210929, −5.51371389440024600874353815428, −4.27573403289110511579746909248, −2.50548082726046503084787330654, −1.40440037418246502783140319046, 0.34741861146254621023266694094, 1.74499977385893939013498684005, 4.28308309591632679812180780757, 5.53089621229289724387822279608, 6.21817706539320160633525455319, 7.35472501731497717831276935419, 7.77804208274432324055714921916, 8.809675831070374203134177699497, 9.884347459254466222095796503699, 10.53383781817151553450821459340

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