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Properties

Label 2-525-35.33-c1-0-16
Degree $2$
Conductor $525$
Sign $0.759 + 0.650i$
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  + (0.679 + 2.53i)2-s + (−0.965 − 0.258i)3-s + (−4.23 + 2.44i)4-s − 2.62i·6-s + (−2.44 − 1.00i)7-s + (−5.35 − 5.35i)8-s + (0.866 + 0.499i)9-s + (−1.94 − 3.36i)11-s + (4.71 − 1.26i)12-s + (0.707 − 0.707i)13-s + (0.891 − 6.88i)14-s + (5.04 − 8.73i)16-s + (−1.34 + 5.02i)17-s + (−0.679 + 2.53i)18-s + (3.33 − 5.78i)19-s + ⋯
L(s)  = 1  + (0.480 + 1.79i)2-s + (−0.557 − 0.149i)3-s + (−2.11 + 1.22i)4-s − 1.07i·6-s + (−0.924 − 0.380i)7-s + (−1.89 − 1.89i)8-s + (0.288 + 0.166i)9-s + (−0.585 − 1.01i)11-s + (1.36 − 0.364i)12-s + (0.196 − 0.196i)13-s + (0.238 − 1.83i)14-s + (1.26 − 2.18i)16-s + (−0.326 + 1.21i)17-s + (−0.160 + 0.597i)18-s + (0.765 − 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.759 + 0.650i$
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: \((2,\ 525,\ (\ :1/2),\ 0.759 + 0.650i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.182712 - 0.0675728i\)
\(L(\frac12)\) \(\approx\) \(0.182712 - 0.0675728i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 + 1.00i)T \)
good2 \( 1 + (-0.679 - 2.53i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.94 + 3.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \)
17 \( 1 + (1.34 - 5.02i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.33 + 5.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.20 - 1.66i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.76iT - 29T^{2} \)
31 \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0133 - 0.0497i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 + (4.97 + 4.97i)T + 43iT^{2} \)
47 \( 1 + (2.04 - 0.547i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.540 + 2.01i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.61 - 6.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.47 + 0.396i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.64T + 71T^{2} \)
73 \( 1 + (9.08 + 2.43i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.22 + 5.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.49 + 1.49i)T - 83iT^{2} \)
89 \( 1 + (7.46 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.34 + 3.34i)T + 97iT^{2} \)
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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

−10.67089730331250581331607424969, −9.641156208182509750581255657934, −8.605159605004823011808757853688, −7.79978406519198698864538791357, −6.92258741269170114389608684642, −6.08720465164456377466598456022, −5.59808807670327860879320222457, −4.36791739805926741172475619621, −3.34503146157089871808641222018, −0.10248654286790987563387213875, 1.80265329249215595436512037591, 3.00765884349617643453648451978, 4.02608252153939603365146937344, 5.06723919883995111947472249147, 5.87715482163002916061695516790, 7.27486577741175171634621315118, 8.846123111321136093495186289473, 9.836030926975996166563075863624, 10.03009082530453043354328280697, 11.06995360083369925314836165261

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