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Properties

Label 2-525-35.24-c2-0-43
Degree $2$
Conductor $525$
Sign $-0.409 + 0.912i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.04 − 1.76i)2-s + (0.866 − 1.5i)3-s + (4.19 − 7.26i)4-s − 6.09i·6-s + (6.99 − 0.244i)7-s − 15.4i·8-s + (−1.5 − 2.59i)9-s + (−1.29 + 2.24i)11-s + (−7.26 − 12.5i)12-s + 11.5·13-s + (20.8 − 13.0i)14-s + (−10.4 − 18.0i)16-s + (11.6 − 20.0i)17-s + (−9.14 − 5.28i)18-s + (−25.9 + 14.9i)19-s + ⋯
L(s)  = 1  + (1.52 − 0.880i)2-s + (0.288 − 0.5i)3-s + (1.04 − 1.81i)4-s − 1.01i·6-s + (0.999 − 0.0348i)7-s − 1.93i·8-s + (−0.166 − 0.288i)9-s + (−0.117 + 0.204i)11-s + (−0.605 − 1.04i)12-s + 0.890·13-s + (1.49 − 0.932i)14-s + (−0.652 − 1.12i)16-s + (0.682 − 1.18i)17-s + (−0.508 − 0.293i)18-s + (−1.36 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.409 + 0.912i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.989959446\)
\(L(\frac12)\) \(\approx\) \(4.989959446\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-6.99 + 0.244i)T \)
good2 \( 1 + (-3.04 + 1.76i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (1.29 - 2.24i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 11.5T + 169T^{2} \)
17 \( 1 + (-11.6 + 20.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (25.9 - 14.9i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (30.4 - 17.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 24.4T + 841T^{2} \)
31 \( 1 + (32.4 + 18.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (22.2 - 12.8i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 3.71iT - 1.68e3T^{2} \)
43 \( 1 - 74.2iT - 1.84e3T^{2} \)
47 \( 1 + (1.68 + 2.92i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-34.6 - 20.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-42.7 - 24.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.765 - 0.441i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (56.3 + 32.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 86.0T + 5.04e3T^{2} \)
73 \( 1 + (-30.7 + 53.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-13.7 - 23.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 131.T + 6.88e3T^{2} \)
89 \( 1 + (-56.5 + 32.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 42.2T + 9.40e3T^{2} \)
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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

−10.76037328811234778679408678249, −9.853971017933104936078078158015, −8.475698135566518395321326105229, −7.58594774559689115806118394957, −6.29695426612577982218988355332, −5.50685384616277861333144977791, −4.44949290882078156712661409487, −3.56031110033449600225928456933, −2.29563165199719240726526311111, −1.37656166728283188402799753012, 2.17199033186667841161617861022, 3.66729702249150873486427951867, 4.26634324191585020121389664494, 5.29303758641659142062630813306, 6.06294774735349034216824089343, 7.06607368111009834046752105193, 8.348807051008930317502835971557, 8.502493658714726653072180749659, 10.36889800306272720623327716018, 11.01890679472938410301209042693

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