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Properties

Label 2-525-15.8-c1-0-7
Degree $2$
Conductor $525$
Sign $0.868 + 0.496i$
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  + (−1.79 − 1.79i)2-s + (−0.491 + 1.66i)3-s + 4.47i·4-s + (3.87 − 2.10i)6-s + (0.707 − 0.707i)7-s + (4.45 − 4.45i)8-s + (−2.51 − 1.63i)9-s − 1.56i·11-s + (−7.43 − 2.19i)12-s + (−2.21 − 2.21i)13-s − 2.54·14-s − 7.09·16-s + (3.60 + 3.60i)17-s + (1.59 + 7.46i)18-s + 1.68i·19-s + ⋯
L(s)  = 1  + (−1.27 − 1.27i)2-s + (−0.283 + 0.958i)3-s + 2.23i·4-s + (1.58 − 0.859i)6-s + (0.267 − 0.267i)7-s + (1.57 − 1.57i)8-s + (−0.839 − 0.543i)9-s − 0.472i·11-s + (−2.14 − 0.634i)12-s + (−0.615 − 0.615i)13-s − 0.680·14-s − 1.77·16-s + (0.874 + 0.874i)17-s + (0.375 + 1.75i)18-s + 0.385i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.868 + 0.496i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.868 + 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619902 - 0.164697i\)
\(L(\frac12)\) \(\approx\) \(0.619902 - 0.164697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.491 - 1.66i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (1.79 + 1.79i)T + 2iT^{2} \)
11 \( 1 + 1.56iT - 11T^{2} \)
13 \( 1 + (2.21 + 2.21i)T + 13iT^{2} \)
17 \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \)
19 \( 1 - 1.68iT - 19T^{2} \)
23 \( 1 + (-0.995 + 0.995i)T - 23iT^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (0.440 - 0.440i)T - 37iT^{2} \)
41 \( 1 - 6.44iT - 41T^{2} \)
43 \( 1 + (-5.47 - 5.47i)T + 43iT^{2} \)
47 \( 1 + (-3.69 - 3.69i)T + 47iT^{2} \)
53 \( 1 + (-2.83 + 2.83i)T - 53iT^{2} \)
59 \( 1 + 5.54T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + (-3.75 + 3.75i)T - 67iT^{2} \)
71 \( 1 + 3.61iT - 71T^{2} \)
73 \( 1 + (-5.89 - 5.89i)T + 73iT^{2} \)
79 \( 1 + 17.0iT - 79T^{2} \)
83 \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \)
89 \( 1 - 9.40T + 89T^{2} \)
97 \( 1 + (4.39 - 4.39i)T - 97iT^{2} \)
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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

−10.43813416878552193044249994225, −10.22526862982835275717895662836, −9.308170609823339898474246235257, −8.367091505589450882946507339518, −7.79838600591418476169755745578, −6.17657499580964066114691456398, −4.79476583010762708293791330343, −3.62610738137239562616302176282, −2.73248835275547646897003079619, −0.928592873889060889421495769307, 0.874964456220734838023374145107, 2.35077446527450845681041609929, 4.91760912830430285206552319687, 5.70439818221018128223873778460, 6.83232864300312333589839920272, 7.22651504580154973562547676906, 8.109027907110932073363327527419, 8.901507389186640192711916555151, 9.753523059154297700782126481308, 10.67024220694512061716712619674

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