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Properties

Label 2-525-105.32-c1-0-1
Degree $2$
Conductor $525$
Sign $-0.507 - 0.861i$
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  + (−1.26 − 0.340i)2-s + (−1.71 + 0.224i)3-s + (−0.236 − 0.136i)4-s + (2.25 + 0.299i)6-s + (−1.25 − 2.32i)7-s + (2.11 + 2.11i)8-s + (2.89 − 0.769i)9-s + (−3.38 − 1.95i)11-s + (0.436 + 0.181i)12-s + (1.56 − 1.56i)13-s + (0.807 + 3.38i)14-s + (−1.69 − 2.92i)16-s + (−0.693 − 2.58i)17-s + (−3.94 − 0.00887i)18-s + (1.61 − 0.930i)19-s + ⋯
L(s)  = 1  + (−0.897 − 0.240i)2-s + (−0.991 + 0.129i)3-s + (−0.118 − 0.0681i)4-s + (0.921 + 0.122i)6-s + (−0.476 − 0.879i)7-s + (0.746 + 0.746i)8-s + (0.966 − 0.256i)9-s + (−1.01 − 0.588i)11-s + (0.125 + 0.0523i)12-s + (0.434 − 0.434i)13-s + (0.215 + 0.903i)14-s + (−0.422 − 0.731i)16-s + (−0.168 − 0.627i)17-s + (−0.929 − 0.00209i)18-s + (0.369 − 0.213i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0164182 + 0.0287334i\)
\(L(\frac12)\) \(\approx\) \(0.0164182 + 0.0287334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.224i)T \)
5 \( 1 \)
7 \( 1 + (1.25 + 2.32i)T \)
good2 \( 1 + (1.26 + 0.340i)T + (1.73 + i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.56 + 1.56i)T - 13iT^{2} \)
17 \( 1 + (0.693 + 2.58i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.638 - 2.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.513T + 29T^{2} \)
31 \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.77 - 6.60i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (7.60 - 7.60i)T - 43iT^{2} \)
47 \( 1 + (5.10 + 1.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.85 + 0.498i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.259 + 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.74 + 2.34i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (-0.749 - 2.79i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.37 - 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.16 + 9.16i)T + 83iT^{2} \)
89 \( 1 + (5.67 + 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.81 - 6.81i)T + 97iT^{2} \)
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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

−10.98770151515517549426125313269, −10.18844095323779745546833720450, −9.770176724774524655112655212564, −8.570260869223418694056607939261, −7.63052295200400612463576705023, −6.72344561137755391159549281809, −5.48435609248334992385130376928, −4.75162634784746173467128238207, −3.29405962051544674931477958160, −1.21322296145994178824055941784, 0.03402545654969177077035002531, 1.96645081098136233243200980744, 3.90247262713358104372629367387, 5.08973484605004866221722553791, 6.05003463969555754290838445464, 7.01408327430836515958290719592, 7.88368342342860835245618498448, 8.826417963547864424551761197634, 9.732536444114536408903272531505, 10.36960955763724057132790687448

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