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Properties

Label 2-525-35.12-c1-0-2
Degree $2$
Conductor $525$
Sign $-0.153 - 0.988i$
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  + (0.595 − 0.159i)2-s + (0.258 − 0.965i)3-s + (−1.40 + 0.810i)4-s − 0.616i·6-s + (−2.22 + 1.43i)7-s + (−1.57 + 1.57i)8-s + (−0.866 − 0.499i)9-s + (2.27 + 3.94i)11-s + (0.419 + 1.56i)12-s + (−0.0478 − 0.0478i)13-s + (−1.09 + 1.20i)14-s + (0.932 − 1.61i)16-s + (−4.03 − 1.07i)17-s + (−0.595 − 0.159i)18-s + (−3.38 + 5.86i)19-s + ⋯
L(s)  = 1  + (0.420 − 0.112i)2-s + (0.149 − 0.557i)3-s + (−0.701 + 0.405i)4-s − 0.251i·6-s + (−0.840 + 0.542i)7-s + (−0.557 + 0.557i)8-s + (−0.288 − 0.166i)9-s + (0.686 + 1.18i)11-s + (0.121 + 0.451i)12-s + (−0.0132 − 0.0132i)13-s + (−0.292 + 0.323i)14-s + (0.233 − 0.403i)16-s + (−0.977 − 0.261i)17-s + (−0.140 − 0.0376i)18-s + (−0.776 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.153 - 0.988i$
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: \((2,\ 525,\ (\ :1/2),\ -0.153 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.596488 + 0.696531i\)
\(L(\frac12)\) \(\approx\) \(0.596488 + 0.696531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 1.43i)T \)
good2 \( 1 + (-0.595 + 0.159i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-2.27 - 3.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0478 + 0.0478i)T + 13iT^{2} \)
17 \( 1 + (4.03 + 1.07i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.38 - 5.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.05 - 3.94i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.68iT - 29T^{2} \)
31 \( 1 + (2.56 - 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.13 + 1.91i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.956iT - 41T^{2} \)
43 \( 1 + (-3.47 + 3.47i)T - 43iT^{2} \)
47 \( 1 + (2.35 + 8.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (12.1 + 3.24i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.09 - 1.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.569 - 0.329i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.72 - 10.1i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.58T + 71T^{2} \)
73 \( 1 + (1.98 - 7.42i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.84 - 2.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.19 + 7.19i)T + 83iT^{2} \)
89 \( 1 + (-7.86 + 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.60 + 4.60i)T - 97iT^{2} \)
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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

−11.42389611112601860239098864304, −10.01360800124124775141160281232, −9.212756693841770025991196763330, −8.589175947970194153724951984999, −7.42442960221139343689043512075, −6.53576809778148596726790118064, −5.50408683863350032083969268445, −4.30678649221232033824446191562, −3.33859464247876017396827148506, −2.02272656314988079088882123852, 0.45919935381155107574906002995, 2.91736884385880575409553055459, 4.05736404270922521786532446296, 4.64332231105483776173682379369, 6.13625070884223103844706915841, 6.48939466221715194154202899225, 8.134324666884914005020375150533, 9.177053783976008352760169167179, 9.456292502355673387937996701411, 10.71157980052958184645994799107

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