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Properties

Label 2-525-35.12-c1-0-22
Degree $2$
Conductor $525$
Sign $0.388 + 0.921i$
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  + (2.53 − 0.679i)2-s + (0.258 − 0.965i)3-s + (4.23 − 2.44i)4-s − 2.62i·6-s + (−1.00 + 2.44i)7-s + (5.35 − 5.35i)8-s + (−0.866 − 0.499i)9-s + (−1.94 − 3.36i)11-s + (−1.26 − 4.71i)12-s + (0.707 + 0.707i)13-s + (−0.891 + 6.88i)14-s + (5.04 − 8.73i)16-s + (5.02 + 1.34i)17-s + (−2.53 − 0.679i)18-s + (−3.33 + 5.78i)19-s + ⋯
L(s)  = 1  + (1.79 − 0.480i)2-s + (0.149 − 0.557i)3-s + (2.11 − 1.22i)4-s − 1.07i·6-s + (−0.380 + 0.924i)7-s + (1.89 − 1.89i)8-s + (−0.288 − 0.166i)9-s + (−0.585 − 1.01i)11-s + (−0.364 − 1.36i)12-s + (0.196 + 0.196i)13-s + (−0.238 + 1.83i)14-s + (1.26 − 2.18i)16-s + (1.21 + 0.326i)17-s + (−0.597 − 0.160i)18-s + (−0.765 + 1.32i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.28987 - 2.18401i\)
\(L(\frac12)\) \(\approx\) \(3.28987 - 2.18401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.00 - 2.44i)T \)
good2 \( 1 + (-2.53 + 0.679i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.94 + 3.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \)
17 \( 1 + (-5.02 - 1.34i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.33 - 5.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.66 + 6.20i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 7.76iT - 29T^{2} \)
31 \( 1 + (5.56 - 3.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0497 + 0.0133i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.55iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + (-0.547 - 2.04i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.01 - 0.540i)T + (45.8 + 26.5i)T^{2} \)
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 \( 1 + (0.396 - 1.47i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.64T + 71T^{2} \)
73 \( 1 + (-2.43 + 9.08i)T + (-63.2 - 36.5i)T^{2} \)
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

−10.93287488062177779578871148345, −10.32923281787953377791003895126, −8.833440745756746409271121131588, −7.82690135035638005808167525960, −6.45670384594682336186063618030, −5.92600313170482702987275645283, −5.14574006169213371146874178160, −3.67284530544948586589943290347, −2.94045975527092710862906226783, −1.76349000315946134722670635769, 2.46380710351974933734555144274, 3.60456289522428734898747984420, 4.34753554069383722890776656288, 5.23764469009509826808450669755, 6.15454107958198235756869119254, 7.35288796656819308006677545755, 7.69928927778539787370562120854, 9.427917730403299456012698832247, 10.37224549415620551594236521040, 11.21550393674169776082790600762

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