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Properties

Label 2-525-105.89-c1-0-31
Degree $2$
Conductor $525$
Sign $-0.263 + 0.964i$
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  + (0.539 − 0.933i)2-s + (−1.46 − 0.918i)3-s + (0.418 + 0.725i)4-s + (−1.64 + 0.876i)6-s + (−0.929 − 2.47i)7-s + 3.05·8-s + (1.31 + 2.69i)9-s + (3.84 − 2.21i)11-s + (0.0513 − 1.44i)12-s − 0.955·13-s + (−2.81 − 0.467i)14-s + (0.812 − 1.40i)16-s + (0.439 − 0.253i)17-s + (3.22 + 0.228i)18-s + (−4.41 − 2.54i)19-s + ⋯
L(s)  = 1  + (0.381 − 0.660i)2-s + (−0.847 − 0.530i)3-s + (0.209 + 0.362i)4-s + (−0.673 + 0.357i)6-s + (−0.351 − 0.936i)7-s + 1.08·8-s + (0.437 + 0.899i)9-s + (1.15 − 0.669i)11-s + (0.0148 − 0.418i)12-s − 0.265·13-s + (−0.752 − 0.125i)14-s + (0.203 − 0.351i)16-s + (0.106 − 0.0615i)17-s + (0.760 + 0.0539i)18-s + (−1.01 − 0.584i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.263 + 0.964i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.884253 - 1.15854i\)
\(L(\frac12)\) \(\approx\) \(0.884253 - 1.15854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.46 + 0.918i)T \)
5 \( 1 \)
7 \( 1 + (0.929 + 2.47i)T \)
good2 \( 1 + (-0.539 + 0.933i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-3.84 + 2.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.955T + 13T^{2} \)
17 \( 1 + (-0.439 + 0.253i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.41 + 2.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.14 + 3.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.89iT - 29T^{2} \)
31 \( 1 + (-5.10 + 2.94i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.51 + 3.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.65T + 41T^{2} \)
43 \( 1 - 0.492iT - 43T^{2} \)
47 \( 1 + (-5.76 - 3.32i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.56 - 7.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.81 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.399 - 0.230i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 - 1.85i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 + (-3.15 - 5.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.38 + 12.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 + (-3.57 + 6.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.91T + 97T^{2} \)
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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

−10.74840132782737004077684930656, −10.24420653605710261412377978049, −8.804652856364655295083344554684, −7.65523212419922826482308049898, −6.85763848493406115195606954361, −6.15324105884675616990229792800, −4.60626016213918595936476287579, −3.88849723630060169985630274281, −2.44104008240702619781872913263, −0.913089128559564419405041608863, 1.68626428707863088681843289657, 3.66793510738813093285796766496, 4.86188116421719040707242831505, 5.51021936201585234619698073027, 6.57753015573444007342990639899, 6.88038322237146714865031614423, 8.508523610994249103584651260463, 9.525627731441502055770003302908, 10.20194846287377811402262548219, 11.11044540834016429935569657234

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