Properties

Label 2-525-35.17-c1-0-5
Degree $2$
Conductor $525$
Sign $-0.452 - 0.891i$
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  + (−0.259 + 0.969i)2-s + (−0.965 + 0.258i)3-s + (0.859 + 0.496i)4-s − 1.00i·6-s + (1.06 − 2.42i)7-s + (−2.12 + 2.12i)8-s + (0.866 − 0.499i)9-s + (−1.78 + 3.08i)11-s + (−0.958 − 0.256i)12-s + (2.78 + 2.78i)13-s + (2.07 + 1.65i)14-s + (−0.514 − 0.891i)16-s + (0.135 + 0.506i)17-s + (0.259 + 0.969i)18-s + (2.06 + 3.57i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.685i)2-s + (−0.557 + 0.149i)3-s + (0.429 + 0.248i)4-s − 0.409i·6-s + (0.401 − 0.915i)7-s + (−0.750 + 0.750i)8-s + (0.288 − 0.166i)9-s + (−0.537 + 0.931i)11-s + (−0.276 − 0.0741i)12-s + (0.772 + 0.772i)13-s + (0.554 + 0.443i)14-s + (−0.128 − 0.222i)16-s + (0.0329 + 0.122i)17-s + (0.0612 + 0.228i)18-s + (0.473 + 0.819i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625784 + 1.01879i\)
\(L(\frac12)\) \(\approx\) \(0.625784 + 1.01879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 2.42i)T \)
good2 \( 1 + (0.259 - 0.969i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (1.78 - 3.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.78 - 2.78i)T + 13iT^{2} \)
17 \( 1 + (-0.135 - 0.506i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.06 - 3.57i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.49 - 0.668i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.14iT - 29T^{2} \)
31 \( 1 + (1.71 + 0.988i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0409 + 0.152i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.28iT - 41T^{2} \)
43 \( 1 + (9.01 - 9.01i)T - 43iT^{2} \)
47 \( 1 + (-5.19 - 1.39i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.48 - 5.55i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.30 - 2.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.67 + 5.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.32 - 1.42i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.23T + 71T^{2} \)
73 \( 1 + (-14.8 + 3.98i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-13.1 + 7.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.42 + 9.42i)T + 83iT^{2} \)
89 \( 1 + (-5.52 - 9.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.48 + 2.48i)T - 97iT^{2} \)
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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

−11.05575511957968531206393020761, −10.43580839577217731755774821947, −9.323526545915361967910334947569, −8.223586889173187438833620564824, −7.35210356486107514705700302078, −6.79677532231762024562820685666, −5.71999514154193588142192759124, −4.69153559607178379340219796102, −3.49629037629950585440529224969, −1.69600912448547546309212986254, 0.827497635282749631470069341720, 2.34906500676949441231517626274, 3.37343858910376367475928723635, 5.17327193501978900586295133642, 5.82868595169046041660178064874, 6.74795500514971140678200523115, 8.026460479207935937297528311561, 8.887468362682738287576488488886, 9.939980945069811217981048231740, 10.82255320385257032735729869377

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