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Properties

Label 2-525-35.33-c1-0-3
Degree $2$
Conductor $525$
Sign $0.997 + 0.0672i$
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  + (−0.448 − 1.67i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.500i)4-s + 1.73i·6-s + (1.15 + 2.38i)7-s + (−1.22 − 1.22i)8-s + (0.866 + 0.499i)9-s + (3 + 5.19i)11-s + (0.965 − 0.258i)12-s + (−3.53 + 3.53i)13-s + (3.46 − 3i)14-s + (−2.5 + 4.33i)16-s + (0.448 − 1.67i)18-s + (−2.59 + 4.5i)19-s + (−0.500 − 2.59i)21-s + (7.34 − 7.34i)22-s + ⋯
L(s)  = 1  + (−0.316 − 1.18i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.250i)4-s + 0.707i·6-s + (0.436 + 0.899i)7-s + (−0.433 − 0.433i)8-s + (0.288 + 0.166i)9-s + (0.904 + 1.56i)11-s + (0.278 − 0.0747i)12-s + (−0.980 + 0.980i)13-s + (0.925 − 0.801i)14-s + (−0.625 + 1.08i)16-s + (0.105 − 0.394i)18-s + (−0.596 + 1.03i)19-s + (−0.109 − 0.566i)21-s + (1.56 − 1.56i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.997 + 0.0672i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.997 + 0.0672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.912963 - 0.0307233i\)
\(L(\frac12)\) \(\approx\) \(0.912963 - 0.0307233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-1.15 - 2.38i)T \)
good2 \( 1 + (0.448 + 1.67i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.53 - 3.53i)T - 13iT^{2} \)
17 \( 1 + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.59 - 4.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.34 + 0.896i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.24 - 8.36i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + (2.44 + 2.44i)T + 43iT^{2} \)
47 \( 1 + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.46 + 6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.36 - 2.24i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-6.76 - 1.81i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.52 + 5.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + (6.92 - 12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.53 + 3.53i)T + 97iT^{2} \)
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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

−11.01173849054282560842591845041, −9.902713277418168474407791544557, −9.528444129303360947238463934620, −8.479330482460841109940529930129, −7.08382019140850421601005315459, −6.36380792894916286348845416941, −4.98351457434379590191039388497, −4.05857514119660203103532196890, −2.34799350776812745547464371721, −1.63085288925035106435446659270, 0.65429552965986658083067366466, 3.06149483509510289380138289946, 4.54282839908702363575908794225, 5.50011033327274458092157324982, 6.41423561422856905989459606606, 7.15643145347060101428117954610, 8.039784620982479616624659103099, 8.843730755124142343842535073897, 9.862446218020885908981991464450, 11.08546329033055657127295491590

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