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Properties

Label 2-525-15.2-c1-0-1
Degree $2$
Conductor $525$
Sign $0.0985 + 0.995i$
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 + (0.394 + 1.68i)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 + (8.81 − 2.06i)12-s + (−3.48 + 3.48i)13-s + 2.68·14-s − 12.8·16-s + (0.131 − 0.131i)17-s + (2.57 − 7.64i)18-s − 3.89i·19-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (0.227 + 0.973i)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 + (2.54 − 0.595i)12-s + (−0.965 + 0.965i)13-s + 0.718·14-s − 3.21·16-s + (0.0319 − 0.0319i)17-s + (0.608 − 1.80i)18-s − 0.893i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.0985 + 0.995i$
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: \((2,\ 525,\ (\ :1/2),\ 0.0985 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154777 - 0.140199i\)
\(L(\frac12)\) \(\approx\) \(0.154777 - 0.140199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.394 - 1.68i)T \)
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

−10.92481942686637836062848890598, −10.22385327377603282632352347619, −9.424192943050683667001424742541, −9.145610484593170439282642910506, −7.954144887019328212457817183179, −7.19278194652392447807463199265, −6.35364772647944748120711069154, −5.10584355352019834150920706381, −4.36963133082711811360570918311, −2.18988114351013465926113519438, 0.17887395890602185205943570818, 1.61782883877744779093315280382, 2.81380227001659173399838607356, 3.54339113672865805132128150275, 5.65619366494692150928551752110, 6.96805065255098913195081019552, 8.067162032167092257024286357532, 8.268369984361282058200836717567, 9.418576180452848032760072927674, 10.10264110454470358836630424824

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