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Properties

Label 2-525-105.59-c1-0-9
Degree $2$
Conductor $525$
Sign $0.968 - 0.248i$
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  + (−0.450 − 0.780i)2-s + (0.248 + 1.71i)3-s + (0.593 − 1.02i)4-s + (1.22 − 0.966i)6-s + (−2.64 + 0.105i)7-s − 2.87·8-s + (−2.87 + 0.853i)9-s + (5.46 + 3.15i)11-s + (1.91 + 0.761i)12-s + 3.77·13-s + (1.27 + 2.01i)14-s + (0.107 + 0.185i)16-s + (3.27 + 1.88i)17-s + (1.96 + 1.86i)18-s + (4.57 − 2.64i)19-s + ⋯
L(s)  = 1  + (−0.318 − 0.551i)2-s + (0.143 + 0.989i)3-s + (0.296 − 0.514i)4-s + (0.500 − 0.394i)6-s + (−0.999 + 0.0398i)7-s − 1.01·8-s + (−0.958 + 0.284i)9-s + (1.64 + 0.950i)11-s + (0.551 + 0.219i)12-s + 1.04·13-s + (0.340 + 0.538i)14-s + (0.0268 + 0.0464i)16-s + (0.793 + 0.458i)17-s + (0.462 + 0.438i)18-s + (1.05 − 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31791 + 0.166173i\)
\(L(\frac12)\) \(\approx\) \(1.31791 + 0.166173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.248 - 1.71i)T \)
5 \( 1 \)
7 \( 1 + (2.64 - 0.105i)T \)
good2 \( 1 + (0.450 + 0.780i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-5.46 - 3.15i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + (-3.27 - 1.88i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.57 + 2.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.38 - 5.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.06iT - 29T^{2} \)
31 \( 1 + (-0.349 - 0.201i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.15 + 0.668i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 + 4.84iT - 43T^{2} \)
47 \( 1 + (-1.68 + 0.970i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.720 + 1.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.60 + 2.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 - 6.56i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.68 + 1.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.21iT - 71T^{2} \)
73 \( 1 + (-1.25 + 2.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.05 - 5.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 + (-0.590 - 1.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.10T + 97T^{2} \)
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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

−10.73884177802221406383226131164, −9.907402628154231739845409744411, −9.354067193244253632041187235059, −8.843206415016231643733144550346, −7.13800589269434021948059668049, −6.21207430242178853916582551908, −5.30248426590107747180678189060, −3.81030288329668635046269033100, −3.12772089594474729263096245265, −1.39337292471728951262843334653, 1.02077082305840425548899022266, 3.00269838822076012113246334073, 3.62522513960802159140724224749, 5.91192731168109635519415383653, 6.37019653494450917527640614114, 7.10050550245467789176350977747, 8.110477357603766014967147871610, 8.836711449504825451417819551312, 9.532138119046446083730192212533, 11.07211758661196925771855217999

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