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Properties

Label 2-525-35.3-c1-0-10
Degree $2$
Conductor $525$
Sign $-0.266 + 0.963i$
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  + (−1.72 − 0.461i)2-s + (−0.258 − 0.965i)3-s + (1.01 + 0.587i)4-s + 1.78i·6-s + (−1.68 + 2.04i)7-s + (1.03 + 1.03i)8-s + (−0.866 + 0.499i)9-s + (1.46 − 2.54i)11-s + (0.304 − 1.13i)12-s + (0.187 − 0.187i)13-s + (3.83 − 2.74i)14-s + (−2.48 − 4.30i)16-s + (3.24 − 0.868i)17-s + (1.72 − 0.461i)18-s + (1.81 + 3.14i)19-s + ⋯
L(s)  = 1  + (−1.21 − 0.326i)2-s + (−0.149 − 0.557i)3-s + (0.508 + 0.293i)4-s + 0.727i·6-s + (−0.635 + 0.772i)7-s + (0.367 + 0.367i)8-s + (−0.288 + 0.166i)9-s + (0.442 − 0.766i)11-s + (0.0877 − 0.327i)12-s + (0.0521 − 0.0521i)13-s + (1.02 − 0.732i)14-s + (−0.621 − 1.07i)16-s + (0.786 − 0.210i)17-s + (0.405 − 0.108i)18-s + (0.417 + 0.722i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341486 - 0.448752i\)
\(L(\frac12)\) \(\approx\) \(0.341486 - 0.448752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.68 - 2.04i)T \)
good2 \( 1 + (1.72 + 0.461i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-1.46 + 2.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.187 + 0.187i)T - 13iT^{2} \)
17 \( 1 + (-3.24 + 0.868i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.81 - 3.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.43 + 9.07i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 0.815iT - 29T^{2} \)
31 \( 1 + (3.76 + 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.31 + 1.69i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 + (-3.59 - 3.59i)T + 43iT^{2} \)
47 \( 1 + (-2.47 + 9.24i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.87 - 1.30i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.41 + 9.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.07 + 4.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.10 + 15.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + (0.508 + 1.89i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.44 - 2.56i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.09 - 6.09i)T - 83iT^{2} \)
89 \( 1 + (4.87 + 8.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.93 - 5.93i)T + 97iT^{2} \)
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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

−10.48992359724098722857961062527, −9.630510024299889836840154621015, −8.797886836427403518534336881511, −8.238775153282998680918379489397, −7.15660408012651569215363602929, −6.15510960389584314282077086012, −5.18306034429975008828971416022, −3.34805100576351439564400584522, −2.08238125595697943466468506729, −0.59239697295597916791543902034, 1.23721917822868069739359398699, 3.38781586317128913621541894409, 4.37304797313968613682728878398, 5.70330175107235585156601059141, 7.08847323827224049264285512390, 7.36174315005534262178258902803, 8.670792308117325587273803765648, 9.513098891801385417756756009003, 9.906628286562520261600521297535, 10.74618855698143401914920513770

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