Properties

Label 2-525-35.3-c1-0-17
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: $\chi_{525} (493, \cdot )$
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

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

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