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Properties

Label 2-525-15.2-c1-0-33
Degree $2$
Conductor $525$
Sign $-0.935 - 0.353i$
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  + (1.90 − 1.90i)2-s + (−1.68 − 0.394i)3-s − 5.22i·4-s + (−3.95 + 2.45i)6-s + (−0.707 − 0.707i)7-s + (−6.13 − 6.13i)8-s + (2.68 + 1.33i)9-s − 3.76i·11-s + (−2.06 + 8.81i)12-s + (−3.48 + 3.48i)13-s − 2.68·14-s − 12.8·16-s + (−0.131 + 0.131i)17-s + (7.64 − 2.57i)18-s − 3.89i·19-s + ⋯
L(s)  = 1  + (1.34 − 1.34i)2-s + (−0.973 − 0.227i)3-s − 2.61i·4-s + (−1.61 + 1.00i)6-s + (−0.267 − 0.267i)7-s + (−2.16 − 2.16i)8-s + (0.896 + 0.443i)9-s − 1.13i·11-s + (−0.595 + 2.54i)12-s + (−0.965 + 0.965i)13-s − 0.718·14-s − 3.21·16-s + (−0.0319 + 0.0319i)17-s + (1.80 − 0.608i)18-s − 0.893i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.935 - 0.353i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.935 - 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.311877 + 1.70880i\)
\(L(\frac12)\) \(\approx\) \(0.311877 + 1.70880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 + 0.394i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-1.90 + 1.90i)T - 2iT^{2} \)
11 \( 1 + 3.76iT - 11T^{2} \)
13 \( 1 + (3.48 - 3.48i)T - 13iT^{2} \)
17 \( 1 + (0.131 - 0.131i)T - 17iT^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + (-3.35 - 3.35i)T + 23iT^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 + (4.98 + 4.98i)T + 37iT^{2} \)
41 \( 1 - 1.16iT - 41T^{2} \)
43 \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \)
47 \( 1 + (-7.97 + 7.97i)T - 47iT^{2} \)
53 \( 1 + (3.80 + 3.80i)T + 53iT^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 + 4.11T + 61T^{2} \)
67 \( 1 + (-0.153 - 0.153i)T + 67iT^{2} \)
71 \( 1 + 2.12iT - 71T^{2} \)
73 \( 1 + (-9.79 + 9.79i)T - 73iT^{2} \)
79 \( 1 + 0.147iT - 79T^{2} \)
83 \( 1 + (-2.58 - 2.58i)T + 83iT^{2} \)
89 \( 1 + 1.17T + 89T^{2} \)
97 \( 1 + (1.52 + 1.52i)T + 97iT^{2} \)
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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

−10.79055326791941523138414427413, −10.00019664697042368572586707172, −9.055721137216577702300533080590, −7.12099719468273807249986200313, −6.32746436020034225312011175692, −5.33537053417681525346095771815, −4.63176260094985715752263050715, −3.54398158724780404821288537667, −2.25085064223077355487828633082, −0.74773542033508907979720793639, 2.86070156259727568788427048359, 4.25288710839373740573218894013, 4.94570450070001331536155347263, 5.70789218591626251898804731619, 6.62571300362827006906150103534, 7.29051792174118497568660283688, 8.222186907172267865729445508739, 9.602365853337292633807314501465, 10.54779855709852229176311080949, 11.89920397911549797526085535174

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